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On second-order fuzzy discrete population model

On second-order fuzzy discrete population model 1IntroductionThe discrete time population model is the most appropriate mathematical description of life histories of organism. These models are used widely in fisheries and many organisms [1]. The Beverton-Holt model also known as the Skellam equation [2] is one of the classic population model that has been studied (1)xn=βxn−11+δxn−1,n=0,1,…,{x}_{n}=\frac{\beta {x}_{n-1}}{1+\delta {x}_{n-1}},\hspace{1em}n=0,1,\ldots ,where xn{x}_{n}is population at the nth generation, β\beta represents a productivity parameter, and δ\delta controls the level of density dependence. Since then, many results on the model and the generation of the model have been widely obtained by some researchers [3,4,5].In model (1), population is assumed to respond instantly to size variations. But in fact, there is a lag between the variations of external conditions and response of the population to these variations. Therefore, population dynamics is indeed described by delay models. For example, Pielou [6] studied the difference equation with delay: (2)xn=αxn−11+βxn−k,n=1,2,…,{x}_{n}=\frac{\alpha {x}_{n-1}}{1+\beta {x}_{n-k}},\hspace{1em}n=1,2,\ldots ,where α>1,β>0\alpha \gt 1,\beta \gt 0and k∈{1,2,…}k\in \left\{1,2,\ldots \right\}.Also the generalization of (2) with many delays (3)xn=αxn−11+∑i=1sβixn−ki,n=1,2,…,{x}_{n}=\frac{\alpha {x}_{n-1}}{1+\mathop{\sum }\limits_{i=1}^{s}{\beta }_{i}{x}_{n-{k}_{i}}},\hspace{1em}n=1,2,\ldots ,where α>1\alpha \gt 1and βi>0{\beta }_{i}\gt 0is studied in [7].The generalization of (2) with infinite memory (4)xn=αxn−11+xn−1+β∑j=1∞cjxn−j,n=1,2,…,{x}_{n}=\frac{\alpha {x}_{n-1}}{1+{x}_{n-1}+\beta \mathop{\sum }\limits_{j=1}^{\infty }{c}_{j}{x}_{n-j}},\hspace{1em}n=1,2,\ldots ,where α>1,β>0\alpha \gt 1,\beta \gt 0and ∑j=1∞cj=1{\sum }_{j=1}^{\infty }{c}_{j}=1, is studied in [8].In fact, the identification of the population dynamics model is usually based on the statistical method, starting from data experimentally obtained and on the choice of some method adapted to the identification. These models, even the classic deterministic approach, are subjected to inaccuracies (fuzzy uncertainty) that can be caused by either the nature of the state variables or by parameters as model coefficients.In our real life, we have learned to deal with uncertainty. Scientists also accept the fact that uncertainty is a very important factor in most applications. Modeling the real life problems in such cases usually involves vagueness or uncertainty. The concept of fuzzy set and system was introduced by Zadeh [9], and its development has been growing rapidly to various situations of theory and application including fuzzy differential and fuzzy difference equations. It is well known that fuzzy difference equation is a difference equation whose parameters or the state variable are fuzzy numbers, and its solutions are sequences of fuzzy numbers. It has been used to model a dynamical system under possibility uncertainty. Due to the applicability of fuzzy difference equation for the analysis of phenomena where imprecision is inherent, this class of difference equation is a very important topic from theoretical point of view and also its applications. Recently, there has been an increasing interest in the study of fuzzy difference equations [10,11,12, 13,14,15, 16,17,18, 19,20,21, 22,23,24, 25,26,27, 28,29].Inspired with the previous publication, by virtue of the theory of fuzzy difference equation, in this work, we consider the following discrete population model with fuzzy state variable: (5)xn=Axn−11+xn−1+Bxn−2,n=1,2,…,{x}_{n}=\frac{A{x}_{n-1}}{1+{x}_{n-1}+B{x}_{n-2}},\hspace{1em}n=1,2,\ldots ,where xn{x}_{n}is the population size at the observation instant nth generation and xn{x}_{n}is a fuzzy number. Parameter AAis regarded as the natural growth coefficient and Al,α>1,α∈(0,1]{A}_{l,\alpha }\gt 1,\alpha \in (0,1]. The variation in the distributive coefficient BB, which is a positive fuzzy number, defines the response of the environment to population growth, depending on the age structure and prehistory of the population.The main aim of this work is to study the existence of positive solutions of the population dynamics model (5). Furthermore, according to a generation of division (gg-division) of fuzzy numbers, we derive some conditions so that every positive solution of population dynamics model (5) is bounded and persistent. Finally, under some conditions, we prove that the population dynamics model (5) has a unique positive equilibrium xxand every positive solution tends to xxas n→∞n\to \infty .2Preliminary and definitionsFirst, we provide the following definitions.Definition 2.1[30] u:R→[0,1]u:R\to \left[0,1]is said to be a fuzzy number if it satisfies conditions (i)–(iv) as follows: (i)uuis normal, i.e., there exists an x∈Rx\in Rsuch that u(x)=1u\left(x)=1;(ii)uuis fuzzy convex, i.e., for all t∈[0,1]t\in \left[0,1]and x1,x2∈R{x}_{1},{x}_{2}\in Rsuch that u(tx1+(1−t)x2)≥min{u(x1),u(x2)};u\left(t{x}_{1}+\left(1-t){x}_{2})\ge \min \left\{u\left({x}_{1}),u\left({x}_{2})\right\};(iii)uuis upper semicontinuous;(iv)The support of uu, suppu=⋃α∈(0,1][u]α¯={x:u(x)>0}¯\hspace{0.1em}\text{supp}\hspace{0.1em}u=\overline{{\bigcup }_{\alpha \in (0,1]}{\left[u]}^{\alpha }}=\overline{\left\{x:u\left(x)\gt 0\right\}}is compact.For α∈(0,1]\alpha \in (0,1], the α\alpha -cuts of fuzzy number uuis [u]α={x∈R:u(x)≥α}{\left[u]}^{\alpha }=\left\{x\in R:u\left(x)\ge \alpha \right\}, and for α=0\alpha =0, the support of uuis defined as suppu=[u]0={x∈R∣u(x)>0}¯{\rm{supp}}\hspace{0.25em}u={\left[u]}^{0}=\overline{\left\{x\in R| u\left(x)\gt 0\right\}}.Definition 2.2Fuzzy number (parametric form) [30] A fuzzy number uuin a parametric form is a pair (u̲,u¯)\left(\underline{u},\overline{u})of functions u̲(r),u¯(r),0≤r≤1\underline{u}\left(r),\overline{u}\left(r),0\le r\le 1, which satisfies the following requirements: (1)u̲(r)\underline{u}\left(r)is a bounded monotonic increasing left continuous function,(2)u¯(r)\overline{u}\left(r)is a bounded monotonic decreasing left continuous function,(3)u̲(r)≤u¯(r),0≤r≤1\underline{u}\left(r)\le \overline{u}\left(r),0\le r\le 1.A crisp (real) number xxis simply represented by (u̲(r),u¯(r))=(x,x),0≤r≤1\left(\underline{u}\left(r),\overline{u}\left(r))=\left(x,x),0\le r\le 1. The fuzzy number space {(u̲(r),u¯(r))}\left\{\left(\underline{u}\left(r),\overline{u}\left(r))\right\}becomes a convex cone E1{E}^{1}, which could be embedded isomorphically and isometrically into a Banach space [30].Definition 2.3[30] The distance between two arbitrary fuzzy numbers uuand vvis defined as follows: (6)D(u,v)=supα∈[0,1]max{∣ul,α−vl,α∣,∣ur,α−vr,α∣}.D\left(u,v)=\mathop{\sup }\limits_{\alpha \in \left[0,1]}{\rm{\max }}\left\{| {u}_{l,\alpha }-{v}_{l,\alpha }| ,| {u}_{r,\alpha }-{v}_{r,\alpha }| \right\}.It is clear that (E1,D)\left({E}^{1},D)is a complete metric space.Definition 2.4[30] Let u=(u̲(r),u¯(r)),v=(v̲(r),v¯(r))∈E1,0≤r≤1,u=\left(\underline{u}\left(r),\overline{u}\left(r)),v=\left(\underline{v}\left(r),\overline{v}\left(r))\in {E}^{1},0\le r\le 1,and arbitrary k∈Rk\in R. Then, (i)u=vu=viff u̲(r)=v̲(r),u¯(r)=v¯(r)\underline{u}\left(r)=\underline{v}\left(r),\overline{u}\left(r)=\overline{v}\left(r),(ii)u+v=(u̲(r)+v̲(r),u¯(r)+v¯(r))u+v=\left(\underline{u}\left(r)+\underline{v}\left(r),\overline{u}\left(r)+\overline{v}\left(r)),(iii)u−v=(u̲(r)−v¯(r),u¯(r)−v̲(r))u-v=\left(\underline{u}\left(r)-\overline{v}\left(r),\overline{u}\left(r)-\underline{v}\left(r)),(iv)ku=(ku̲(r),ku¯(r)),k≥0;(ku¯(r),ku̲(r)),k<0,ku=\left\{\begin{array}{ll}\left(k\underline{u}\left(r),k\overline{u}\left(r)),& k\ge 0;\\ \left(k\overline{u}\left(r),k\underline{u}\left(r)),& k\lt 0,\end{array}\right.(v)uv=(min{u̲(r)v̲(r),u̲(r)v¯(r),u¯(r)v̲(r),u¯(r)v¯(r)},max{u̲(r)v̲(r),u̲(r)v¯(r),u¯(r)v̲(r),u¯(r)v¯(r)})uv=\left(\min \left\{\underline{u}\left(r)\underline{v}\left(r),\underline{u}\left(r)\overline{v}\left(r),\overline{u}\left(r)\underline{v}\left(r),\overline{u}\left(r)\overline{v}\left(r)\right\},{\rm{\max }}\left\{\underline{u}\left(r)\underline{v}\left(r),\underline{u}\left(r)\overline{v}\left(r),\overline{u}\left(r)\underline{v}\left(r),\overline{u}\left(r)\overline{v}\left(r)\right\}).Definition 2.5(Triangular fuzzy number) [30] A triangular fuzzy number (TFN) denoted by AAis defined as (a,b,c)\left(a,b,c), where the membership function: A(x)=0,x≤a;x−ab−a,a≤x≤b;1,x=b;c−xc−b,b≤x≤c;0,x≥c.A\left(x)=\left\{\begin{array}{ll}\phantom{\rule[-0.5em]{}{0ex}}0,& x\le a;\\ \phantom{\rule[-1.25em]{}{0ex}}\frac{x-a}{b-a},& a\le x\le b;\\ \phantom{\rule[-0.5em]{}{0ex}}1,& x=b;\\ \phantom{\rule[-1em]{}{0ex}}\frac{c-x}{c-b},& b\le x\le c;\\ 0,& x\ge c.\end{array}\right.The α\alpha -cuts of A=(a,b,c)A=\left(a,b,c)are described by [A]α={x∈R:A(x)≥α}=[a+α(b−a),c−α(c−b)]=[Al,α,Ar,α]{\left[A]}^{\alpha }=\left\{x\in R:A\left(x)\ge \alpha \right\}=\left[a+\alpha \left(b-a),c-\alpha \left(c-b)]=\left[{A}_{l,\alpha },{A}_{r,\alpha }], α∈[0,1]\alpha \in \left[0,1], and it is clear that [A]α{\left[A]}^{\alpha }are a closed interval. A fuzzy number AAis positive if suppA⊂(0,∞)\hspace{0.1em}\text{supp}\hspace{0.1em}A\subset \left(0,\infty ).The following proposition is fundamental since it characterizes a fuzzy set through the α\alpha -levels.Proposition 2.1[30] If {Aα:α∈[0,1]}\left\{{A}^{\alpha }:\alpha \in \left[0,1]\right\}is a compact, convex, and not empty subset family of Rn{R}^{n}such that(i)⋃Aα¯⊂A0\overline{\bigcup {A}^{\alpha }}\subset {A}^{0}.(ii)Aα2⊂Aα1{A}^{{\alpha }_{2}}\subset {A}^{{\alpha }_{1}}if α1≤α2{\alpha }_{1}\le {\alpha }_{2}.(iii)Aα=⋂k≥1Aαk{A}^{\alpha }={\bigcap }_{k\ge 1}{A}^{{\alpha }_{k}}if αk↑α>0{\alpha }_{k}\uparrow \alpha \gt 0.Then, there is u∈Enu\in {E}^{n}(En{E}^{n}denotes n dimensional fuzzy number space) such that [u]α=Aα{\left[u]}^{\alpha }={A}^{\alpha }for all α∈(0,1]\alpha \in (0,1]and [u]0=⋃0<α≤1Aα¯⊂A0{\left[u]}^{0}=\overline{{\bigcup }_{0\lt \alpha \le 1}{A}^{\alpha }}\subset {A}^{0}.Definition 2.6[31] Suppose that A,B∈E1A,B\in {E}^{1}have α\alpha -cuts [A]α=[Al,α,Ar,α],[B]α=[Bl,α,Br,α]{\left[A]}^{\alpha }=\left[{A}_{l,\alpha },{A}_{r,\alpha }],{\left[B]}^{\alpha }=\left[{B}_{l,\alpha },{B}_{r,\alpha }], with 0∉[B]α,∀α∈[0,1]0\notin {\left[B]}^{\alpha },\forall \hspace{-0.25em}\alpha \in \left[0,1]. The gg-division ÷g{\div}_{g}is the operation that calculates the fuzzy number C=A÷gBC=A{\div}_{g}Bhaving level cuts [C]α=[Cl,α,Cr,α]{\left[C]}^{\alpha }=\left[{C}_{l,\alpha },{C}_{r,\alpha }](here [A]α−1=[1/Ar,α,1/Al,α]{{\left[A]}^{\alpha }}^{-1}=\left[1\hspace{0.1em}\text{/}{A}_{r,\alpha },1\text{/}\hspace{0.1em}{A}_{l,\alpha }]) defined by (7)[C]α=[A]α÷g[B]α⇔(i)[A]α=[B]α[C]α,or(ii)[B]α=[A]α[C]α−1{\left[C]}^{\alpha }={\left[A]}^{\alpha }\hspace{0.25em}{\div}_{g}\hspace{0.25em}{\left[B]}^{\alpha }\hspace{0.33em}\iff \hspace{0.33em}\left\{\begin{array}{ll}\left(i)& {\left[A]}^{\alpha }={\left[B]}^{\alpha }{\left[C]}^{\alpha },\\ {\rm{or}}& \\ \left(ii)& {\left[B]}^{\alpha }={\left[A]}^{\alpha }{{\left[C]}^{\alpha }}^{-1}\end{array}\right.provided that CCis a proper fuzzy number (Cl,α{C}_{l,\alpha }is nondecreasing, Cr,α{C}_{r,\alpha }is nonincreasing, Cl,1≤Cr,1{C}_{l,1}\le {C}_{r,1}).Remark 2.1According to [31], in this paper, the fuzzy number is positive, if A÷gB=C∈E1A{\div}_{g}B=C\in {E}^{1}exists, then the following two cases are possible:Case I. If Al,αBr,α≤Ar,αBl,α,∀α∈[0,1]{A}_{l,\alpha }{B}_{r,\alpha }\le {A}_{r,\alpha }{B}_{l,\alpha },\forall \alpha \in \left[0,1], then Cl,α=Al,αBl,α,Cr,α=Ar,αBr,α{C}_{l,\alpha }=\frac{{A}_{l,\alpha }}{{B}_{l,\alpha }},{C}_{r,\alpha }=\frac{{A}_{r,\alpha }}{Br,\alpha },Case II. If Al,αBr,α≥Ar,αBl,α,∀α∈[0,1]{A}_{l,\alpha }{B}_{r,\alpha }\ge {A}_{r,\alpha }{B}_{l,\alpha },\forall \alpha \in \left[0,1], then Cl,α=Ar,αBr,α,Cr,α=Al,αBl,α{C}_{l,\alpha }=\frac{{A}_{r,\alpha }}{{B}_{r,\alpha }},{C}_{r,\alpha }=\frac{{A}_{l,\alpha }}{Bl,\alpha }.The fuzzy analog of the boundedness and persistence (see [22,23]) is as follows:Definition 2.7A sequence of positive fuzzy numbers (xn)\left({x}_{n})is persistence (resp. bounded) if there exists a positive real number MM(resp. NN) such that suppxn⊂[M,∞)(resp.suppxn⊂(0,N]),n=1,2,…,{\rm{supp}}\hspace{0.33em}{x}_{n}\subset {[}M,\infty )({\rm{resp}}.{\rm{supp}}\hspace{0.33em}{x}_{n}\subset \left(0,N]),\hspace{1em}n=1,2,\ldots ,A sequence of positive fuzzy numbers (xn)\left({x}_{n})is bounded and persistence if there exist positive real numbers M,N>0M,N\gt 0such that suppxn⊂[M,N],n=1,2,…{\rm{supp}}\hspace{0.33em}{x}_{n}\subset \left[M,N],\hspace{1em}n=1,2,\ldots A sequence of positive fuzzy numbers (xn),n=1,2,…\left({x}_{n}),n=1,2,\ldots , is an unbounded if the norm ‖xn‖,n=1,2,…\Vert {x}_{n}\Vert ,n=1,2,\ldots , is an unbounded sequence.Definition 2.8xn{x}_{n}is said to be a positive solution of (5) if (xn)\left({x}_{n})is a sequence of positive fuzzy numbers, which satisfy (5). A positive fuzzy number xxis called a positive equilibrium of (5) if x=Ax1+x+Bx.x=\frac{Ax}{1+x+Bx}.Let (xn)\left({x}_{n})be a sequence of positive fuzzy numbers and xxis a positive fuzzy number, xn→x{x}_{n}\to xas n→∞n\to \infty if limn→∞D(xn,x)=0{\mathrm{lim}}_{n\to \infty }\hspace{0.33em}D\left({x}_{n},x)=0.3Main results3.1Existence of positive solution of population dynamics model (5)First, we study the existence of positive solutions of the population dynamics model (5). We need the following lemma.Lemma 3.1[30] Let f:R+×R+→R+f:{R}^{+}\times {R}^{+}\to {R}^{+}be continuous, and, A,BA,Bare fuzzy numbers. Then, (8)[f(A,B)]α=f([A]α,[B]α)α∈(0,1].{[f\left(A,B)]}^{\alpha }=f\left({\left[A]}^{\alpha },{\left[B]}^{\alpha })\hspace{1em}\alpha \in (0,1].Theorem 3.1If A,BA,B, and initial values x−2,x−1{x}_{-2},{x}_{-1}of population dynamics model (5) are positive fuzzy numbers, then, there exists a unique positive solution xn{x}_{n}of the population dynamics model (5).ProofSuppose that there exists a sequence of fuzzy numbers (xn)\left({x}_{n})satisfying (5) with initial condition x−1,x−2{x}_{-1},{x}_{-2}. Consider the α\alpha -cuts, α∈(0,1]\alpha \in (0,1], (9)[xn]α=[Ln,α,Rn,α],n=0,1,2,…{\left[{x}_{n}]}^{\alpha }=\left[{L}_{n,\alpha },{R}_{n,\alpha }],\hspace{1em}n=0,1,2,\ldots It follows from (5), (9), and Lemma 3.1 that [xn]α=[Ln,α,Rn,α]=Axn−11+xn−1+Bxn−2α=[A]α×[xn−1]α1+[xn−1]α+[B]α×[xn−2]α=[Al,αLn−1,α,Ar,αRn−1,α][1+Ln−1,α+Bl,αLn−2,α,1+Rn−1,α+Br,αRn−2,α].\begin{array}{rcl}{\left[{x}_{n}]}^{\alpha }& =& \left[{L}_{n,\alpha },{R}_{n,\alpha }]\\ & =& {\left[\frac{A{x}_{n-1}}{1+{x}_{n-1}+B{x}_{n-2}}\right]}^{\alpha }\\ & =& \frac{{\left[A]}^{\alpha }\times {\left[{x}_{n-1}]}^{\alpha }}{1+{\left[{x}_{n-1}]}^{\alpha }+{\left[B]}^{\alpha }\times {\left[{x}_{n-2}]}^{\alpha }}\\ & =& \frac{\left[{A}_{l,\alpha }{L}_{n-1,\alpha },{A}_{r,\alpha }{R}_{n-1,\alpha }]}{\left[1+{L}_{n-1,\alpha }+{B}_{l,\alpha }{L}_{n-2,\alpha },1+{R}_{n-1,\alpha }+{B}_{r,\alpha }{R}_{n-2,\alpha }]}.\end{array}Noting Remark 2.1, one of the following two cases holds:Case I: (10)[xn]α=[Ln,α,Rn,α]=Al,αLn−1,α1+Ln−1,α+Bl,αLn−2,α,Ar,αRn−1,α1+Rn−1,α+Br,αRn−2,α.{\left[{x}_{n}]}^{\alpha }=\left[{L}_{n,\alpha },{R}_{n,\alpha }]=\left[\frac{{A}_{l,\alpha }{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+{B}_{l,\alpha }{L}_{n-2,\alpha }},\frac{{A}_{r,\alpha }{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }+{B}_{r,\alpha }{R}_{n-2,\alpha }}\right].Case II: (11)[xn]α=[Ln,α,Rn,α]=Ar,αRn−1,α1+Rn−1,α+Br,αRn−2,α,Al,αLn−1,α1+Ln−1,α+Bl,αLn−2,α.{\left[{x}_{n}]}^{\alpha }=\left[{L}_{n,\alpha },{R}_{n,\alpha }]=\left[\frac{{A}_{r,\alpha }{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }+{B}_{r,\alpha }{R}_{n-2,\alpha }},\frac{{A}_{l,\alpha }{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+{B}_{l,\alpha }{L}_{n-2,\alpha }}\right].If Case I holds true, it follows that for n∈{0,1,2,…},α∈(0,1]n\in \left\{0,1,2,\ldots \right\},\alpha \in (0,1]: (12)Ln,α=Al,αLn−1,α1+Ln−1,α+Bl,αLn−2,α,Rn,α=Ar,αRn−1,α1+Rn−1,α+Br,αRn−2,α.{L}_{n,\alpha }=\frac{{A}_{l,\alpha }{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+{B}_{l,\alpha }{L}_{n-2,\alpha }},\hspace{1em}{R}_{n,\alpha }=\frac{{A}_{r,\alpha }{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }+{B}_{r,\alpha }{R}_{n-2,\alpha }}.□Next, the proof is similar to those of Theorem 3.1 [10]. We omit it.Remark 3.1From theoretical point of view, the existence of solution for fuzzy difference equation is very important with initial condition. Therefore, in this sense, the existence of positive fuzzy solution for discrete population dynamics model is of vital importance and practical significance. In fact, according to Theorem 3.1, the positive solution of discrete population dynamics model with fuzzy state is a sequence of positive fuzzy numbers, which can describe the fuzzy uncertainty of the dynamics model.3.2Dynamics of discrete population model (5)To study the dynamical behavior of the positive solutions of the discrete population model (5), according to Definition 2.6, we consider two cases.First, if Case I holds true, the following lemma is essential to the proof of next theorem.Lemma 3.2Consider the difference equation: (13)yn=kyn−11+yn−1+βyn−2,n=1,2,…,{y}_{n}=\frac{k{y}_{n-1}}{1+{y}_{n-1}+\beta {y}_{n-2}},\hspace{1em}n=1,2,\ldots ,where y−2,y−1∈(0,+∞){y}_{-2},{y}_{-1}\in \left(0,+\infty ), if k>1,0≤β<1k\gt 1,0\le \beta \lt 1, then the following statements are true. (i)The system exists a trivial equilibrium point y∗=0{y}^{\ast }=0, which is unstable.(ii)The system exists a unique positive equilibrium y∗=k−1β+1{y}^{\ast }=\frac{k-1}{\beta +1}, which is globally asymptotically stable.(iii)Every positive solution yn{y}_{n}of (13) is bounded and persistent.Proof (i) Let y∗{y}^{\ast }be an equilibrium point of (13). It is easy to obtain that there exist two equilibrium points: y∗=0,andy∗=k−1β+1.{y}^{\ast }=0,\hspace{1.0em}{\rm{and}}\hspace{1.0em}{y}^{\ast }=\frac{k-1}{\beta +1}.It is clear that the trivial equilibrium point y∗{y}^{\ast }is unstable. So we omit it.(ii) The linearized equation associated with (13) at equilibrium expressed as follows: y∗=k−1β+1{y}^{\ast }=\frac{k-1}{\beta +1}is expressed as follows: yn−kβ+1k(β+1)yn−1+β(k−1)k(β+1)yn−2=0.{y}_{n}-\frac{k\beta +1}{k\left(\beta +1)}{y}_{n-1}+\frac{\beta \left(k-1)}{k\left(\beta +1)}{y}_{n-2}=0.\hspace{1.8em}Since k>1,0≤β<1k\gt 1,0\le \beta \lt 1, it is easy to obtain (14)kβ+1k(β+1)+β(k−1)k(β+1)<1.\frac{k\beta +1}{k\left(\beta +1)}+\frac{\beta \left(k-1)}{k\left(\beta +1)}\lt 1.By virtue of Theorem 1.3.7 in [7], we have that the system is locally asymptotically stable.On the other hand, it is similar to the proof of Theorem 1 in [32], and we can show that limn→∞yn=k−1β+1{\mathrm{lim}}_{n\to \infty }\hspace{0.33em}{y}_{n}=\frac{k-1}{\beta +1}. Thus, the unique positive equilibrium y∗=k−1β+1{y}^{\ast }=\frac{k-1}{\beta +1}is globally asymptotically stable.(iii) Let yn{y}_{n}be a solution of (13). We consider the following difference equation: (15)un=kun−11+un−1,n=1,2,3,…,{u}_{n}=\frac{k{u}_{n-1}}{1+{u}_{n-1}},\hspace{1em}n=1,2,3,\ldots ,and the initial values of (15) are satisfied with (16)y0≤u0.{y}_{0}\le {u}_{0}.It follows from (13), (15), and (16) that (17)yn=kyn−11+yn−1+βyn−2≤kyn−11+yn−1≤kun−11+un−1=un,n≥2.{y}_{n}=\frac{k{y}_{n-1}}{1+{y}_{n-1}+\beta {y}_{n-2}}\le \frac{k{y}_{n-1}}{1+{y}_{n-1}}\le \frac{k{u}_{n-1}}{1+{u}_{n-1}}={u}_{n},\hspace{1em}n\ge 2.It is clear that every solution un{u}_{n}of (15) converges to equilibrium k−1k-1, i.e., limn→∞un=k−1{\mathrm{lim}}_{n\to \infty }\hspace{0.25em}{u}_{n}=k-1. Therefore, it follows from (17) that (18)yn≤limn→∞supyn≤limn→∞supun=limn→∞un=k−1.\hspace{-21.15em}{y}_{n}\le \mathop{\mathrm{lim}}\limits_{n\to \infty }\sup {y}_{n}\le \mathop{\mathrm{lim}}\limits_{n\to \infty }\sup {u}_{n}=\mathop{\mathrm{lim}}\limits_{n\to \infty }{u}_{n}=k-1.On the other hand, we consider the family of sequences {un(ε)}\left\{{u}_{n}\left(\varepsilon )\right\}, where (19)un(ε)=kun−1(ε)1+un−1(ε)+β(k−1+ε),n=2,3,….{u}_{n}\left(\varepsilon )=\frac{k{u}_{n-1}\left(\varepsilon )}{1+{u}_{n-1}\left(\varepsilon )+\beta \left(k-1+\varepsilon )},\hspace{1em}n=2,3,\ldots .Without loss of generality, we take 0<ε<(k−1)(1−β)β0\lt \varepsilon \lt \frac{\left(k-1)\left(1-\beta )}{\beta }. For every fixed ε\varepsilon , difference equation (19) has a stationary trajectory such that u(ε)=limn→∞un(ε)=(k−1)(1−β)−βεu\left(\varepsilon )={\mathrm{lim}}_{n\to \infty }\hspace{0.33em}{u}_{n}\left(\varepsilon )=\left(k-1)\left(1-\beta )-\beta \varepsilon . From (ii), there exists an n0∈N{n}_{0}\in Nsuch that yn≤k−1+ε{y}_{n}\le k-1+\varepsilon for n≥n0n\ge {n}_{0}.Let the initial conditions of (19) be positive and satisfy the conditions: (20)un0+1(ε)≤yn0+1.{u}_{{n}_{0}+1}\left(\varepsilon )\le {y}_{{n}_{0}+1}.Then, we obtain that (21)yn=kyn−11+yn−1+βyn−2≥kyn−11+yn−1+β(k−1+ε)≥kun−1(ε)1+un−1(ε)+β(k−1+ε)=un(ε).{y}_{n}=\frac{k{y}_{n-1}}{1+{y}_{n-1}+\beta {y}_{n-2}}\ge \frac{k{y}_{n-1}}{1+{y}_{n-1}+\beta \left(k-1+\varepsilon )}\ge \frac{k{u}_{n-1}\left(\varepsilon )}{1+{u}_{n-1}\left(\varepsilon )+\beta \left(k-1+\varepsilon )}={u}_{n}\left(\varepsilon ).Therefore, yn≥un(ε){y}_{n}\ge {u}_{n}\left(\varepsilon ), limn→∞infyn≥limn→∞infun(ε)=limn→∞un(ε)=(k−1)(1−β){\mathrm{lim}}_{n\to \infty }\inf {y}_{n}\ge {\mathrm{lim}}_{n\to \infty }\inf {u}_{n}\left(\varepsilon )={\mathrm{lim}}_{n\to \infty }{u}_{n}\left(\varepsilon )=\left(k-1)\left(1-\beta ).So yn≥(k−1)(1−β){y}_{n}\ge \left(k-1)\left(1-\beta )for n≥n0n\ge {n}_{0}. This completes the proof.□Theorem 3.2Consider the discrete population model (5), where parameters AAand BBand the initial conditions x−1,x0{x}_{-1},{x}_{0}are positive fuzzy numbers. If(22)Al,α>1,Br,α<1,α∈(0,1],{A}_{l,\alpha }\gt 1,\hspace{0.33em}{B}_{r,\alpha }\lt 1,\hspace{0.33em}\alpha \in (0,1],then the following statements are true. (i)Every positive solution xn{x}_{n}of discrete fuzzy population model (5) is bounded and persistent.(ii)Every positive solution xn{x}_{n}of discrete fuzzy population model (5) tends to the positive equilibrium point xxas n→∞n\to \infty .Proof (i) Since A,BA,Band the initial value x−1,x0{x}_{-1},{x}_{0}are positive fuzzy numbers, there exist positive real numbers MA,NA,MB,NB,M0,N0,M−1,N−1{M}_{A},{N}_{A},{M}_{B},{N}_{B},{M}_{0},{N}_{0},{M}_{-1},{N}_{-1}such that, for all α∈(0,1]\alpha \in (0,1], (23)[Al,α,Ar,α]⊂[MA,NA],[Bl,α,Br,α]⊂[MB,NB],[L0,α,R0,α]⊂[M0,N0],[L−l,α,R−1,α]⊂[M−1,N−1].\left[{A}_{l,\alpha },{A}_{r,\alpha }]\subset \left[{M}_{A},{N}_{A}],\hspace{1em}\left[{B}_{l,\alpha },{B}_{r,\alpha }]\subset \left[{M}_{B},{N}_{B}],\hspace{1em}\left[{L}_{0,\alpha },{R}_{0,\alpha }]\subset \left[{M}_{0},{N}_{0}],\hspace{1em}\left[{L}_{-l,\alpha },{R}_{-1,\alpha }]\subset \left[{M}_{-1},{N}_{-1}].Let xn{x}_{n}be a positive solution of discrete fuzzy population model (5), from (22), (23), and Lemma 3.2, we obtain (24)Ln,α>(Al,α−1)(1−Bl,α)>(MA−1)(1−NB)≔M,Rn,α<Ar,α−1<NA−1≔N{L}_{n,\alpha }\gt \left({A}_{l,\alpha }-1)\left(1-{B}_{l,\alpha })\gt \left({M}_{A}-1)\left(1-{N}_{B}):= M,\hspace{1em}{R}_{n,\alpha }\lt {A}_{r,\alpha }-1\lt {N}_{A}-1:= NFrom which, we get for n≥2,⋃α∈(0,1][Ln,α,Rn,α]⊂[M,N]n\ge 2,{\bigcup }_{\alpha \in (0,1]}\left[{L}_{n,\alpha },{R}_{n,\alpha }]\subset \left[M,N], and so ⋃α∈(0,1][Ln,α,Rn,α]¯⊆[M,N]\overline{{\bigcup }_{\alpha \in (0,1]}\left[{L}_{n,\alpha },{R}_{n,\alpha }]}\subseteq \left[M,N]. Thus, the positive solution is bounded and persistent.(ii) Suppose that there exists a fuzzy number xxsuch that (25)x=Ax1+x+Bx,[x]α=[Lα,Rα],α∈(0,1].x=\frac{Ax}{1+x+Bx},\hspace{1em}{\left[x]}^{\alpha }=\left[{L}_{\alpha },{R}_{\alpha }],\hspace{1em}\alpha \in (0,1].where Lα,Rα≥0{L}_{\alpha },{R}_{\alpha }\ge 0. Then, from (25), we can prove that (26)Lα=Al,αLα1+Lα+Bl,αLα,Rα=Ar,αRα1+Rα+Br,αRα.{L}_{\alpha }=\frac{{A}_{l,\alpha }{L}_{\alpha }}{1+{L}_{\alpha }+{B}_{l,\alpha }{L}_{\alpha }},\hspace{1em}{R}_{\alpha }=\frac{{A}_{r,\alpha }{R}_{\alpha }}{1+{R}_{\alpha }+{B}_{r,\alpha }{R}_{\alpha }}.\hspace{1.15em}Hence, from (26), we have (27)Lα=Al,α−1Bl,α+1,Rα=Ar,α−1Br,α+1.{L}_{\alpha }=\frac{{A}_{l,\alpha }-1}{{B}_{l,\alpha }+1},\hspace{1em}{R}_{\alpha }=\frac{{A}_{r,\alpha }-1}{{B}_{r,\alpha }+1}.Let xn{x}_{n}be a positive solution of discrete fuzzy population model (5). Since (22) holds true, we can apply Lemma 3.2 to system (13), and so we have (28)limn→∞Ln,α=Lα,limn→∞Rn,α=Rα.\mathop{\mathrm{lim}}\limits_{n\to \infty }{L}_{n,\alpha }={L}_{\alpha },\hspace{1em}\mathop{\mathrm{lim}}\limits_{n\to \infty }{R}_{n,\alpha }={R}_{\alpha }.Therefore, from (28), we have limn→∞D(xn,x)=limn→∞supα∈(0,1]{max{∣Ln,α−Lα∣,∣Rn,α−Rα∣}}=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }D\left({x}_{n},x)=\mathop{\mathrm{lim}}\limits_{n\to \infty }\mathop{\sup }\limits_{\alpha \in (0,1]}\left\{{\rm{\max }}\left\{| {L}_{n,\alpha }-{L}_{\alpha }| ,| {R}_{n,\alpha }-{R}_{\alpha }| \right\}\right\}=0.This completes the proof of the theorem.□Second, if Case II holds true, it follows that for n∈{0,1,2,…},α∈(0,1]n\in \left\{0,1,2,\ldots \right\},\alpha \in (0,1](29)Ln,α=Ar,αRn−1,α1+Rn−1,α+Br,αRn−2,α,Rn,α=Al,αLn−1,α1+Ln−1,α+Bl,αRn−2,α.{L}_{n,\alpha }=\frac{{A}_{r,\alpha }{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }+{B}_{r,\alpha }{R}_{n-2,\alpha }},\hspace{1em}{R}_{n,\alpha }=\frac{{A}_{l,\alpha }{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+{B}_{l,\alpha }{R}_{n-2,\alpha }}.We need the following lemmas.Lemma 3.3Consider the system of difference equations: (30)yn=azn−11+zn−1+czn−2,zn=ayn−11+yn−1+cyn−2,n=1,2,…,{y}_{n}=\frac{a{z}_{n-1}}{1+{z}_{n-1}+c{z}_{n-2}},\hspace{1em}{z}_{n}=\frac{a{y}_{n-1}}{1+{y}_{n-1}+c{y}_{n-2}},\hspace{1em}n=1,2,\ldots ,where a,c,y−1,y0,z−1,z0∈(0,+∞)a,c,{y}_{-1},{y}_{0},{z}_{-1},{z}_{0}\in \left(0,+\infty ). If(31)a>1,a\gt 1,then the following statements are true.(i)The system exists a trivial equilibrium point (0,0)\left(0,0), which is unstable.(ii)The system exists a unique positive equilibrium point (y∗,z∗)=a−1c+1,a−1c+1({y}^{\ast },{z}^{\ast })=\left(\frac{a-1}{c+1},\frac{a-1}{c+1}\right), which is locally asymptotically stable.Proof(i) The linearized equation of system (30) about (0,0)\left(0,0)is expressed as follows: (32)Ψn=GΨn−1,{\Psi }_{n}=G{\Psi }_{n-1},where Ψn−1=yn−1yn−2zn−1zn−2,G=00a01000a0000010.{\Psi }_{n-1}=\left(\begin{array}{c}{y}_{n-1}\\ {y}_{n-2}\\ {z}_{n-1}\\ {z}_{n-2}\end{array}\right),\hspace{1em}G=\left(\begin{array}{cccc}0& 0& a& 0\\ 1& 0& 0& 0\\ a& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right).The characteristic equation of (32) is expressed as follows: (33)λ2(λ2−a2)=0.{\lambda }^{2}\left({\lambda }^{2}-{a}^{2})=0.It follows from (31) that there are two roots of characteristic equation outside the unit disk. So the trivial equilibrium (0,0)\left(0,0)is unstable.(ii) The linearized equation of system (30) about (y∗,z∗)({y}^{\ast },{z}^{\ast })is expressed as follows: (34)Ψn=HΨn−1,{\Psi }_{n}=H{\Psi }_{n-1},where H=00a+acz∗(1+z∗+cz∗)2−acz∗(1+z∗+cz∗)21000a+acy∗(1+y∗+cy∗)2−acy∗(1+y∗+cy∗)2000010.H=\left(\begin{array}{cccc}0& 0& \frac{a+ac{z}^{\ast }}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}& -\frac{ac{z}^{\ast }}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}\\ 1& 0& 0& 0\\ \frac{a+ac{y}^{\ast }}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}& -\frac{ac{y}^{\ast }}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}& 0& 0\\ 0& 0& 1& 0\end{array}\right).Let λ1,λ2,λ3,λ4{\lambda }_{1},{\lambda }_{2},{\lambda }_{3},{\lambda }_{4}denote the eigenvalues of matrix HH, let K=diag(m1,m2,m3,m4)K=\hspace{0.1em}\text{diag}\hspace{0.1em}\left({m}_{1},{m}_{2},{m}_{3},{m}_{4})be a diagonal matrix, where m1=m3=1,mk=1−kε(k=2,4){m}_{1}={m}_{3}=1,{m}_{k}=1-k\varepsilon \left(k=2,4), and (35)0<ε<141−acz∗(1+z∗+cz∗)2−a−acz∗.0\lt \varepsilon \lt \frac{1}{4}\left(1-\frac{ac{z}^{\ast }}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}-a-ac{z}^{\ast }}\right).Clearly, KKis invertible. Computing matrix KHK−1KH{K}^{-1}, we obtain that (36)KHK−1=00(a+acz∗)m1m3−1(1+z∗+cz∗)2−acz∗m1m4−1(1+z∗+cz∗)2m2m1−1000(a+acy∗)m3m1−1(1+y∗+cy∗)2−acy∗m3m2−1(1+y∗+cy∗)20000m4m3−10.KH{K}^{-1}=\left(\begin{array}{cccc}0& 0& \frac{\left(a+ac{z}^{\ast }){m}_{1}{m}_{3}^{-1}}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}& -\frac{ac{z}^{\ast }{m}_{1}{m}_{4}^{-1}}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}\\ {m}_{2}{m}_{1}^{-1}& 0& 0& 0\\ \frac{\left(a+ac{y}^{\ast }){m}_{3}{m}_{1}^{-1}}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}& -\frac{ac{y}^{\ast }{m}_{3}{m}_{2}^{-1}}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}& 0& 0\\ 0& 0& {m}_{4}{m}_{3}^{-1}& 0\end{array}\right).It is obtained from (36) that a+acz∗(1+z∗+cz∗)2+acz∗m4−1(1+z∗+cz∗)2=a+acz∗1−11−4ε(1+z∗+cz∗)2<1\frac{a+ac{z}^{\ast }}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}+\frac{ac{z}^{\ast }{m}_{4}^{-1}}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}=\frac{a+ac{z}^{\ast }\left(1-\frac{1}{1-4\varepsilon }\right)}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}\lt 1a+acy∗(1+y∗+cy∗)2+acy∗m2−1(1+y∗+cy∗)2=a+acy∗1−11−2ε(1+y∗+cy∗)2<1.\frac{a+ac{y}^{\ast }}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}+\frac{ac{y}^{\ast }{m}_{2}^{-1}}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}=\frac{a+ac{y}^{\ast }\left(1-\frac{1}{1-2\varepsilon }\right)}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}\lt 1.It is well known that HHhas the same eigenvalues as KHK−1KH{K}^{-1}, and we have (37)max1≤i≤4∣λi∣≤‖KHK−1‖∞=maxm2m1−1,m4m3−1,a+acz∗+acz∗m1m4−1(1+z∗+cz∗)2,a+acy∗+acy∗m3m2−1(1+y∗+cy∗)2<1.\mathop{\max }\limits_{1\le i\le 4}| {\lambda }_{i}| \le \Vert KH{K}^{-1}{\Vert }_{\infty }=\max \left\{{m}_{2}{m}_{1}^{-1},{m}_{4}{m}_{3}^{-1},\frac{a+ac{z}^{\ast }+ac{z}^{\ast }{m}_{1}{m}_{4}^{-1}}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}},\frac{a+ac{y}^{\ast }+ac{y}^{\ast }{m}_{3}{m}_{2}^{-1}}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}\right\}\lt 1.This implies that the equilibrium (y∗,z∗)({y}^{\ast },{z}^{\ast })of (30) is locally asymptotically stable.□Theorem 3.3Suppose that parameters A and B of discrete fuzzy population model (5) are positive trivial fuzzy numbers (positive real numbers) and A>1,0<B<1A\gt 1,0\lt B\lt 1. If the following condition are satisfied, (38)1+BLn−2,α1+BRn−2,α≤Ln−1,αRn−1,α,n=1,2,…,∀α∈(0,1].\frac{1+B{L}_{n-2,\alpha }}{1+B{R}_{n-2,\alpha }}\le \frac{{L}_{n-1,\alpha }}{{R}_{n-1,\alpha }},\hspace{1em}n=1,2,\ldots ,\forall \alpha \in (0,1].Then, the following statements are true. (i)Every positive solution xn{x}_{n}of discrete fuzzy population model (5) is bounded and persistence.(ii)Every positive solution xn{x}_{n}of discrete fuzzy population model (5) tends to the positive equilibrium point xxas n→+∞n\to +\infty .Proof (i) The proof is similar to those of (i) in Theorem 3.2. Let xn{x}_{n}be a positive solution of discrete fuzzy population model (5). It is easy to get that Ln,α≥(A−1)(1−B),Rn,α≤A−1,∀α∈(0,1].{L}_{n,\alpha }\ge \left(A-1)\left(1-B),\hspace{1em}{R}_{n,\alpha }\le A-1,\hspace{1em}\forall \alpha \in (0,1].From which, we obtain for n≥2,⋃α∈(0,1][Ln,α,Rn,α]⊂[(A−1)(1−B),A−1]n\ge 2,{\bigcup }_{\alpha \in (0,1]}\left[{L}_{n,\alpha },{R}_{n,\alpha }]\subset \left[\left(A-1)\left(1-B),A-1], and so ⋃α∈(0,1][Ln,α,Rn,α]¯⊆[(A−1)(1−B),A−1]\overline{{\bigcup }_{\alpha \in (0,1]}\left[{L}_{n,\alpha },{R}_{n,\alpha }]}\hspace{0.25em}\subseteq \left[\left(A-1)\left(1-B),A-1]. Thus, the positive solution is bounded and persistent.(ii) Suppose that there exists a positive fuzzy number xxsatisfying (25). Then from (25), we have (39)Lα=ARα1+Rα+BRα,Rα=ALα1+Lα+BLα,∀α∈(0,1].{L}_{\alpha }=\frac{A{R}_{\alpha }}{1+{R}_{\alpha }+B{R}_{\alpha }},\hspace{1em}{R}_{\alpha }=\frac{A{L}_{\alpha }}{1+{L}_{\alpha }+B{L}_{\alpha }},\forall \alpha \in (0,1].From (39), we have Lα=Rα=A−1B+1.{L}_{\alpha }={R}_{\alpha }=\frac{A-1}{B+1}.Let xn{x}_{n}be a positive solution of discrete fuzzy population model (5) such that Case II holds. Namely, (40)Ln,α=ARn−1,α1+Rn−1,α++BRn−2,α,Rn,α=ALn−1,α1+Ln−1,α+BLn−2,α.{L}_{n,\alpha }=\frac{A{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }++B{R}_{n-2,\alpha }},\hspace{1em}{R}_{n,\alpha }=\frac{A{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+B{L}_{n-2,\alpha }}.Since A>1,0<B<1A\gt 1,0\lt B\lt 1and (38) is satisfied, we can apply Lemma 3.3 to system (40), and so we have (41)limn→∞Ln,α=Lα,limn→∞Rn,α=Rα.\mathop{\mathrm{lim}}\limits_{n\to \infty }{L}_{n,\alpha }={L}_{\alpha },\hspace{1em}\mathop{\mathrm{lim}}\limits_{n\to \infty }{R}_{n,\alpha }={R}_{\alpha }.Therefore, from (41), we have limn→∞D(xn,x)=limn→∞supα∈(0,1]{max{∣Ln,α−Lα∣,∣Rn,α−Rα∣}}=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }D\left({x}_{n},x)=\mathop{\mathrm{lim}}\limits_{n\to \infty }\mathop{\sup }\limits_{\alpha \in (0,1]}\left\{{\rm{\max }}\left\{| {L}_{n,\alpha }-{L}_{\alpha }| ,| {R}_{n,\alpha }-{R}_{\alpha }| \right\}\right\}=0.This completes the proof of Theorem 3.3.□Remark 3.2In the population dynamical model, the parameters of model are derived from statistic data with vagueness or uncertainty. It corresponds to reality to use fuzzy parameters in the population dynamical model. In contrast with the classic population model, the solution of fuzzy population model is within a range of value (approximate value), which are taken into account fuzzy uncertainties. Furthermore, the global asymptotic behavior of the discrete second-order population model are obtained in fuzzy context.4Numerical examplesExample 4.1Consider the following second-order fuzzy discrete population model: (42)xn=Axn−11+xn−1+Bxn−2,n=1,2,…,{x}_{n}=\frac{A{x}_{n-1}}{1+{x}_{n-1}+B{x}_{n-2}},\hspace{1em}n=1,2,\ldots ,and we take A,B,A,B,and the initial values x−1,x0{x}_{-1},{x}_{0}such that (43)A(x)=2x−3,1.5≤x≤2−2x+5,2≤x≤2.5,x−1(x)=2x−1,0.5≤x≤1−5x+6,1≤x≤1.2\begin{array}{ll}A\left(x)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}2x-3,& 1.5\le x\le 2\\ -2x+5,& 2\le x\le 2.5\end{array}\right.,& {x}_{-1}\left(x)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}2x-1,& 0.5\le x\le 1\\ -5x+6,& 1\le x\le 1.2\end{array}\right.\end{array}(44)B(x)=5x−2,0.4≤x≤0.6−5x+4,0.6≤x≤0.8,x0(x)=x−3,3≤x≤4−x+5,4≤x≤5.\begin{array}{ll}B\left(x)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}5x-2,& 0.4\le x\le 0.6\\ -5x+4,& 0.6\le x\le 0.8\end{array}\right.,& {x}_{0}\left(x)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}x-3,& 3\le x\le 4\\ -x+5,& 4\le x\le 5.\end{array}\right.\end{array}From (43), we obtain (45)[A]α=1.5+12α,2.5−12α,[x−1]α=0.5+12α,1.2−15α,α∈(0,1].{\left[A]}^{\alpha }=\left[1.5+\frac{1}{2}\alpha ,2.5-\frac{1}{2}\alpha \right],\hspace{1em}{\left[{x}_{-1}]}^{\alpha }=\left[0.5+\frac{1}{2}\alpha ,1.2-\frac{1}{5}\alpha \right],\hspace{1em}\alpha \in (0,1].From (44), we obtain (46)[B]α=0.4+15α,0.8−15α,[x0]α=[3+α,5−α],α∈(0,1].{\left[B]}^{\alpha }=\left[0.4+\frac{1}{5}\alpha ,0.8-\frac{1}{5}\alpha \right],\hspace{1em}{\left[{x}_{0}]}^{\alpha }={[}3+\alpha ,5-\alpha ],\hspace{1em}\alpha \in (0,1].Therefore, it follows that (47)⋃α∈(0,1][A]α¯=[1.5,2.5],⋃α∈(0,1][x−1]α¯=[0.5,1.2],⋃α∈(0,1][B]α¯=[0.4,0.8],⋃α∈(0,1][x0]α¯=[3,5].\overline{\bigcup _{\alpha \in (0,1]}{\left[A]}^{\alpha }}=\left[1.5,2.5],\hspace{1em}\overline{\bigcup _{\alpha \in (0,1]}{\left[{x}_{-1}]}^{\alpha }}=\left[0.5,1.2],\hspace{1em}\overline{\bigcup _{\alpha \in (0,1]}{\left[B]}^{\alpha }}=\left[0.4,0.8],\hspace{1em}\overline{\bigcup _{\alpha \in (0,1]}{\left[{x}_{0}]}^{\alpha }}=\left[3,5].From (42), it results in a coupled system of difference equations with parameter α\alpha , (48)Ln,α=Al,αLn−1,α1+Ln−1,α+Bl,αLn−1,α,Rn,α=Ar,αRn−1,α1+Rn−1,α+Br,αRn−1,α,α∈(0,1].{L}_{n,\alpha }=\frac{{A}_{l,\alpha }{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+{B}_{l,\alpha }{L}_{n-1,\alpha }},\hspace{1em}{R}_{n,\alpha }=\frac{{A}_{r,\alpha }{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }+{B}_{r,\alpha }{R}_{n-1,\alpha }},\hspace{1em}\alpha \in (0,1].Therefore, Al,α>1,Br,α<1,∀α∈(0,1]{A}_{l,\alpha }\gt 1,{B}_{r,\alpha }\lt 1,\forall \alpha \in (0,1], and initial values x0{x}_{0}are positive fuzzy numbers, so from Theorem 3.2, we have that every positive solution xn{x}_{n}of equation (42) is bounded and persistence. In addition, from Theorem 3.2, equation (42) has a unique positive equilibrium x¯=(0.357,0.625,0.833)\overline{x}=\left(0.357,0.625,0.833). Moreover, every positive solution xn{x}_{n}of equation (42) converges the unique equilibrium x¯\overline{x}with respect to DDas n→∞n\to \infty (see Figures 1, 2, 3).Figure 1The dynamics of system (42).Figure 2The solution of system (48) at α=0\alpha =0and α=0.25\alpha =0.25.Figure 3The solution of system (48) at α=0.75\alpha =0.75and α=1\alpha =1.Example 4.2Consider the second-order fuzzy discrete population model (42), where A=1.5,B=0.6A=1.5,B=0.6, and the initial values x0,x−1{x}_{0},{x}_{-1}are satisfied (49)x−1(x)=0.5x−0.5,1≤x≤3−0.5x+2.5,3≤x≤5,x0(x)=2x−6,3≤x≤3.5−2x+8,3.5≤x≤4.{x}_{-1}\left(x)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}0.5x-0.5,& 1\le x\le 3\\ -0.5x+2.5,& 3\le x\le 5\end{array}\right.\hspace{0.33em},\hspace{1em}{x}_{0}\left(x)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}2x-6,& 3\le x\le 3.5\\ -2x+8,& 3.5\le x\le 4.\end{array}\right.From (48), we obtain (50)[x−1]α=[1+2α,5−2α],[x0]α=3+12α,4−12α,α∈(0,1].{\left[{x}_{-1}]}^{\alpha }={[}1+2\alpha ,5-2\alpha ],\hspace{1em}{\left[{x}_{0}]}^{\alpha }=\left[3+\frac{1}{2}\alpha ,4-\frac{1}{2}\alpha \right],\hspace{1em}\alpha \in (0,1].Therefore, it follows that (51)⋃α∈(0,1][x−1]α¯=[1,5],⋃α∈(0,1][x0]α¯=[3,4].\overline{\bigcup _{\alpha \in (0,1]}{\left[{x}_{-1}]}^{\alpha }}=\left[1,5],\hspace{1em}\overline{\bigcup _{\alpha \in (0,1]}{\left[{x}_{0}]}^{\alpha }}=\left[3,4].From (42), it results in a coupled system of difference equation with parameter α\alpha , (52)Ln,α=ARn−1,α1+Rn−1,α+BRn−2,α,Rn,α=ALn−1,α1+Ln−1,α+BLn−2,α,α∈(0,1].{L}_{n,\alpha }=\frac{A{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }+B{R}_{n-2,\alpha }},\hspace{1em}{R}_{n,\alpha }=\frac{A{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+B{L}_{n-2,\alpha }},\hspace{1em}\alpha \in (0,1].It is clear that (38) is satisfied and initial values x−1,x0{x}_{-1},{x}_{0}are positive fuzzy numbers, so from Theorem 3.3, equation (42) has a unique positive equilibrium x¯=0.3125\overline{x}=0.3125. Moreover, every positive solution xn{x}_{n}of equation (42) converges the unique equilibrium x¯\overline{x}with respect to DDas n→∞n\to \infty (see Figures 4, 5, 6).Figure 4The dynamics of system (42).Figure 5The solution of system (48) at α=0\alpha =0and α=0.25\alpha =0.25.Figure 6The solution of system (48) at α=0.75\alpha =0.75and α=1\alpha =1.5ConclusionIn this work, according to a generalization of division (gg-division) of fuzzy number, we study the second-order fuzzy discrete population model xn=Axn−11+xn−1+Bxn−2{x}_{n}=\frac{A{x}_{n-1}}{1+{x}_{n-1}+B{x}_{n-2}}. The existence of the positive solution and the qualitative behavior to (5) are investigated. The main results are as follows: (1)Under Case I, the positive solution is bounded and persistent if Al,α>1,Br,α<1,α∈(0,1]{A}_{l,\alpha }\gt 1,{B}_{r,\alpha }\lt 1,\alpha \in (0,1]. Every positive solution xn{x}_{n}tends to the unique equilibrium xxas n→∞n\to \infty .(2)Under Case II, the positive solution is bounded and persistent if A,BA,Bare positive trivial fuzzy numbers and A>1,B<1,1+BLn−2,α1+BRn−2,α≤Ln−1,αRn−1,α,n=1,2,…,∀α∈(0,1]A\gt 1,B\lt 1,\frac{1+B{L}_{n-2,\alpha }}{1+B{R}_{n-2,\alpha }}\le \frac{{L}_{n-1,\alpha }}{{R}_{n-1,\alpha }},\hspace{0.33em}n=1,2,\ldots ,\forall \alpha \in (0,1]. Every positive solution xn{x}_{n}tends to the unique equilibrium xxas n→∞n\to \infty . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

On second-order fuzzy discrete population model

Open Mathematics , Volume 20 (1): 15 – Jan 1, 2022

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de Gruyter
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© 2022 Qianhong Zhang et al., published by De Gruyter
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2391-5455
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2391-5455
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10.1515/math-2022-0018
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Abstract

1IntroductionThe discrete time population model is the most appropriate mathematical description of life histories of organism. These models are used widely in fisheries and many organisms [1]. The Beverton-Holt model also known as the Skellam equation [2] is one of the classic population model that has been studied (1)xn=βxn−11+δxn−1,n=0,1,…,{x}_{n}=\frac{\beta {x}_{n-1}}{1+\delta {x}_{n-1}},\hspace{1em}n=0,1,\ldots ,where xn{x}_{n}is population at the nth generation, β\beta represents a productivity parameter, and δ\delta controls the level of density dependence. Since then, many results on the model and the generation of the model have been widely obtained by some researchers [3,4,5].In model (1), population is assumed to respond instantly to size variations. But in fact, there is a lag between the variations of external conditions and response of the population to these variations. Therefore, population dynamics is indeed described by delay models. For example, Pielou [6] studied the difference equation with delay: (2)xn=αxn−11+βxn−k,n=1,2,…,{x}_{n}=\frac{\alpha {x}_{n-1}}{1+\beta {x}_{n-k}},\hspace{1em}n=1,2,\ldots ,where α>1,β>0\alpha \gt 1,\beta \gt 0and k∈{1,2,…}k\in \left\{1,2,\ldots \right\}.Also the generalization of (2) with many delays (3)xn=αxn−11+∑i=1sβixn−ki,n=1,2,…,{x}_{n}=\frac{\alpha {x}_{n-1}}{1+\mathop{\sum }\limits_{i=1}^{s}{\beta }_{i}{x}_{n-{k}_{i}}},\hspace{1em}n=1,2,\ldots ,where α>1\alpha \gt 1and βi>0{\beta }_{i}\gt 0is studied in [7].The generalization of (2) with infinite memory (4)xn=αxn−11+xn−1+β∑j=1∞cjxn−j,n=1,2,…,{x}_{n}=\frac{\alpha {x}_{n-1}}{1+{x}_{n-1}+\beta \mathop{\sum }\limits_{j=1}^{\infty }{c}_{j}{x}_{n-j}},\hspace{1em}n=1,2,\ldots ,where α>1,β>0\alpha \gt 1,\beta \gt 0and ∑j=1∞cj=1{\sum }_{j=1}^{\infty }{c}_{j}=1, is studied in [8].In fact, the identification of the population dynamics model is usually based on the statistical method, starting from data experimentally obtained and on the choice of some method adapted to the identification. These models, even the classic deterministic approach, are subjected to inaccuracies (fuzzy uncertainty) that can be caused by either the nature of the state variables or by parameters as model coefficients.In our real life, we have learned to deal with uncertainty. Scientists also accept the fact that uncertainty is a very important factor in most applications. Modeling the real life problems in such cases usually involves vagueness or uncertainty. The concept of fuzzy set and system was introduced by Zadeh [9], and its development has been growing rapidly to various situations of theory and application including fuzzy differential and fuzzy difference equations. It is well known that fuzzy difference equation is a difference equation whose parameters or the state variable are fuzzy numbers, and its solutions are sequences of fuzzy numbers. It has been used to model a dynamical system under possibility uncertainty. Due to the applicability of fuzzy difference equation for the analysis of phenomena where imprecision is inherent, this class of difference equation is a very important topic from theoretical point of view and also its applications. Recently, there has been an increasing interest in the study of fuzzy difference equations [10,11,12, 13,14,15, 16,17,18, 19,20,21, 22,23,24, 25,26,27, 28,29].Inspired with the previous publication, by virtue of the theory of fuzzy difference equation, in this work, we consider the following discrete population model with fuzzy state variable: (5)xn=Axn−11+xn−1+Bxn−2,n=1,2,…,{x}_{n}=\frac{A{x}_{n-1}}{1+{x}_{n-1}+B{x}_{n-2}},\hspace{1em}n=1,2,\ldots ,where xn{x}_{n}is the population size at the observation instant nth generation and xn{x}_{n}is a fuzzy number. Parameter AAis regarded as the natural growth coefficient and Al,α>1,α∈(0,1]{A}_{l,\alpha }\gt 1,\alpha \in (0,1]. The variation in the distributive coefficient BB, which is a positive fuzzy number, defines the response of the environment to population growth, depending on the age structure and prehistory of the population.The main aim of this work is to study the existence of positive solutions of the population dynamics model (5). Furthermore, according to a generation of division (gg-division) of fuzzy numbers, we derive some conditions so that every positive solution of population dynamics model (5) is bounded and persistent. Finally, under some conditions, we prove that the population dynamics model (5) has a unique positive equilibrium xxand every positive solution tends to xxas n→∞n\to \infty .2Preliminary and definitionsFirst, we provide the following definitions.Definition 2.1[30] u:R→[0,1]u:R\to \left[0,1]is said to be a fuzzy number if it satisfies conditions (i)–(iv) as follows: (i)uuis normal, i.e., there exists an x∈Rx\in Rsuch that u(x)=1u\left(x)=1;(ii)uuis fuzzy convex, i.e., for all t∈[0,1]t\in \left[0,1]and x1,x2∈R{x}_{1},{x}_{2}\in Rsuch that u(tx1+(1−t)x2)≥min{u(x1),u(x2)};u\left(t{x}_{1}+\left(1-t){x}_{2})\ge \min \left\{u\left({x}_{1}),u\left({x}_{2})\right\};(iii)uuis upper semicontinuous;(iv)The support of uu, suppu=⋃α∈(0,1][u]α¯={x:u(x)>0}¯\hspace{0.1em}\text{supp}\hspace{0.1em}u=\overline{{\bigcup }_{\alpha \in (0,1]}{\left[u]}^{\alpha }}=\overline{\left\{x:u\left(x)\gt 0\right\}}is compact.For α∈(0,1]\alpha \in (0,1], the α\alpha -cuts of fuzzy number uuis [u]α={x∈R:u(x)≥α}{\left[u]}^{\alpha }=\left\{x\in R:u\left(x)\ge \alpha \right\}, and for α=0\alpha =0, the support of uuis defined as suppu=[u]0={x∈R∣u(x)>0}¯{\rm{supp}}\hspace{0.25em}u={\left[u]}^{0}=\overline{\left\{x\in R| u\left(x)\gt 0\right\}}.Definition 2.2Fuzzy number (parametric form) [30] A fuzzy number uuin a parametric form is a pair (u̲,u¯)\left(\underline{u},\overline{u})of functions u̲(r),u¯(r),0≤r≤1\underline{u}\left(r),\overline{u}\left(r),0\le r\le 1, which satisfies the following requirements: (1)u̲(r)\underline{u}\left(r)is a bounded monotonic increasing left continuous function,(2)u¯(r)\overline{u}\left(r)is a bounded monotonic decreasing left continuous function,(3)u̲(r)≤u¯(r),0≤r≤1\underline{u}\left(r)\le \overline{u}\left(r),0\le r\le 1.A crisp (real) number xxis simply represented by (u̲(r),u¯(r))=(x,x),0≤r≤1\left(\underline{u}\left(r),\overline{u}\left(r))=\left(x,x),0\le r\le 1. The fuzzy number space {(u̲(r),u¯(r))}\left\{\left(\underline{u}\left(r),\overline{u}\left(r))\right\}becomes a convex cone E1{E}^{1}, which could be embedded isomorphically and isometrically into a Banach space [30].Definition 2.3[30] The distance between two arbitrary fuzzy numbers uuand vvis defined as follows: (6)D(u,v)=supα∈[0,1]max{∣ul,α−vl,α∣,∣ur,α−vr,α∣}.D\left(u,v)=\mathop{\sup }\limits_{\alpha \in \left[0,1]}{\rm{\max }}\left\{| {u}_{l,\alpha }-{v}_{l,\alpha }| ,| {u}_{r,\alpha }-{v}_{r,\alpha }| \right\}.It is clear that (E1,D)\left({E}^{1},D)is a complete metric space.Definition 2.4[30] Let u=(u̲(r),u¯(r)),v=(v̲(r),v¯(r))∈E1,0≤r≤1,u=\left(\underline{u}\left(r),\overline{u}\left(r)),v=\left(\underline{v}\left(r),\overline{v}\left(r))\in {E}^{1},0\le r\le 1,and arbitrary k∈Rk\in R. Then, (i)u=vu=viff u̲(r)=v̲(r),u¯(r)=v¯(r)\underline{u}\left(r)=\underline{v}\left(r),\overline{u}\left(r)=\overline{v}\left(r),(ii)u+v=(u̲(r)+v̲(r),u¯(r)+v¯(r))u+v=\left(\underline{u}\left(r)+\underline{v}\left(r),\overline{u}\left(r)+\overline{v}\left(r)),(iii)u−v=(u̲(r)−v¯(r),u¯(r)−v̲(r))u-v=\left(\underline{u}\left(r)-\overline{v}\left(r),\overline{u}\left(r)-\underline{v}\left(r)),(iv)ku=(ku̲(r),ku¯(r)),k≥0;(ku¯(r),ku̲(r)),k<0,ku=\left\{\begin{array}{ll}\left(k\underline{u}\left(r),k\overline{u}\left(r)),& k\ge 0;\\ \left(k\overline{u}\left(r),k\underline{u}\left(r)),& k\lt 0,\end{array}\right.(v)uv=(min{u̲(r)v̲(r),u̲(r)v¯(r),u¯(r)v̲(r),u¯(r)v¯(r)},max{u̲(r)v̲(r),u̲(r)v¯(r),u¯(r)v̲(r),u¯(r)v¯(r)})uv=\left(\min \left\{\underline{u}\left(r)\underline{v}\left(r),\underline{u}\left(r)\overline{v}\left(r),\overline{u}\left(r)\underline{v}\left(r),\overline{u}\left(r)\overline{v}\left(r)\right\},{\rm{\max }}\left\{\underline{u}\left(r)\underline{v}\left(r),\underline{u}\left(r)\overline{v}\left(r),\overline{u}\left(r)\underline{v}\left(r),\overline{u}\left(r)\overline{v}\left(r)\right\}).Definition 2.5(Triangular fuzzy number) [30] A triangular fuzzy number (TFN) denoted by AAis defined as (a,b,c)\left(a,b,c), where the membership function: A(x)=0,x≤a;x−ab−a,a≤x≤b;1,x=b;c−xc−b,b≤x≤c;0,x≥c.A\left(x)=\left\{\begin{array}{ll}\phantom{\rule[-0.5em]{}{0ex}}0,& x\le a;\\ \phantom{\rule[-1.25em]{}{0ex}}\frac{x-a}{b-a},& a\le x\le b;\\ \phantom{\rule[-0.5em]{}{0ex}}1,& x=b;\\ \phantom{\rule[-1em]{}{0ex}}\frac{c-x}{c-b},& b\le x\le c;\\ 0,& x\ge c.\end{array}\right.The α\alpha -cuts of A=(a,b,c)A=\left(a,b,c)are described by [A]α={x∈R:A(x)≥α}=[a+α(b−a),c−α(c−b)]=[Al,α,Ar,α]{\left[A]}^{\alpha }=\left\{x\in R:A\left(x)\ge \alpha \right\}=\left[a+\alpha \left(b-a),c-\alpha \left(c-b)]=\left[{A}_{l,\alpha },{A}_{r,\alpha }], α∈[0,1]\alpha \in \left[0,1], and it is clear that [A]α{\left[A]}^{\alpha }are a closed interval. A fuzzy number AAis positive if suppA⊂(0,∞)\hspace{0.1em}\text{supp}\hspace{0.1em}A\subset \left(0,\infty ).The following proposition is fundamental since it characterizes a fuzzy set through the α\alpha -levels.Proposition 2.1[30] If {Aα:α∈[0,1]}\left\{{A}^{\alpha }:\alpha \in \left[0,1]\right\}is a compact, convex, and not empty subset family of Rn{R}^{n}such that(i)⋃Aα¯⊂A0\overline{\bigcup {A}^{\alpha }}\subset {A}^{0}.(ii)Aα2⊂Aα1{A}^{{\alpha }_{2}}\subset {A}^{{\alpha }_{1}}if α1≤α2{\alpha }_{1}\le {\alpha }_{2}.(iii)Aα=⋂k≥1Aαk{A}^{\alpha }={\bigcap }_{k\ge 1}{A}^{{\alpha }_{k}}if αk↑α>0{\alpha }_{k}\uparrow \alpha \gt 0.Then, there is u∈Enu\in {E}^{n}(En{E}^{n}denotes n dimensional fuzzy number space) such that [u]α=Aα{\left[u]}^{\alpha }={A}^{\alpha }for all α∈(0,1]\alpha \in (0,1]and [u]0=⋃0<α≤1Aα¯⊂A0{\left[u]}^{0}=\overline{{\bigcup }_{0\lt \alpha \le 1}{A}^{\alpha }}\subset {A}^{0}.Definition 2.6[31] Suppose that A,B∈E1A,B\in {E}^{1}have α\alpha -cuts [A]α=[Al,α,Ar,α],[B]α=[Bl,α,Br,α]{\left[A]}^{\alpha }=\left[{A}_{l,\alpha },{A}_{r,\alpha }],{\left[B]}^{\alpha }=\left[{B}_{l,\alpha },{B}_{r,\alpha }], with 0∉[B]α,∀α∈[0,1]0\notin {\left[B]}^{\alpha },\forall \hspace{-0.25em}\alpha \in \left[0,1]. The gg-division ÷g{\div}_{g}is the operation that calculates the fuzzy number C=A÷gBC=A{\div}_{g}Bhaving level cuts [C]α=[Cl,α,Cr,α]{\left[C]}^{\alpha }=\left[{C}_{l,\alpha },{C}_{r,\alpha }](here [A]α−1=[1/Ar,α,1/Al,α]{{\left[A]}^{\alpha }}^{-1}=\left[1\hspace{0.1em}\text{/}{A}_{r,\alpha },1\text{/}\hspace{0.1em}{A}_{l,\alpha }]) defined by (7)[C]α=[A]α÷g[B]α⇔(i)[A]α=[B]α[C]α,or(ii)[B]α=[A]α[C]α−1{\left[C]}^{\alpha }={\left[A]}^{\alpha }\hspace{0.25em}{\div}_{g}\hspace{0.25em}{\left[B]}^{\alpha }\hspace{0.33em}\iff \hspace{0.33em}\left\{\begin{array}{ll}\left(i)& {\left[A]}^{\alpha }={\left[B]}^{\alpha }{\left[C]}^{\alpha },\\ {\rm{or}}& \\ \left(ii)& {\left[B]}^{\alpha }={\left[A]}^{\alpha }{{\left[C]}^{\alpha }}^{-1}\end{array}\right.provided that CCis a proper fuzzy number (Cl,α{C}_{l,\alpha }is nondecreasing, Cr,α{C}_{r,\alpha }is nonincreasing, Cl,1≤Cr,1{C}_{l,1}\le {C}_{r,1}).Remark 2.1According to [31], in this paper, the fuzzy number is positive, if A÷gB=C∈E1A{\div}_{g}B=C\in {E}^{1}exists, then the following two cases are possible:Case I. If Al,αBr,α≤Ar,αBl,α,∀α∈[0,1]{A}_{l,\alpha }{B}_{r,\alpha }\le {A}_{r,\alpha }{B}_{l,\alpha },\forall \alpha \in \left[0,1], then Cl,α=Al,αBl,α,Cr,α=Ar,αBr,α{C}_{l,\alpha }=\frac{{A}_{l,\alpha }}{{B}_{l,\alpha }},{C}_{r,\alpha }=\frac{{A}_{r,\alpha }}{Br,\alpha },Case II. If Al,αBr,α≥Ar,αBl,α,∀α∈[0,1]{A}_{l,\alpha }{B}_{r,\alpha }\ge {A}_{r,\alpha }{B}_{l,\alpha },\forall \alpha \in \left[0,1], then Cl,α=Ar,αBr,α,Cr,α=Al,αBl,α{C}_{l,\alpha }=\frac{{A}_{r,\alpha }}{{B}_{r,\alpha }},{C}_{r,\alpha }=\frac{{A}_{l,\alpha }}{Bl,\alpha }.The fuzzy analog of the boundedness and persistence (see [22,23]) is as follows:Definition 2.7A sequence of positive fuzzy numbers (xn)\left({x}_{n})is persistence (resp. bounded) if there exists a positive real number MM(resp. NN) such that suppxn⊂[M,∞)(resp.suppxn⊂(0,N]),n=1,2,…,{\rm{supp}}\hspace{0.33em}{x}_{n}\subset {[}M,\infty )({\rm{resp}}.{\rm{supp}}\hspace{0.33em}{x}_{n}\subset \left(0,N]),\hspace{1em}n=1,2,\ldots ,A sequence of positive fuzzy numbers (xn)\left({x}_{n})is bounded and persistence if there exist positive real numbers M,N>0M,N\gt 0such that suppxn⊂[M,N],n=1,2,…{\rm{supp}}\hspace{0.33em}{x}_{n}\subset \left[M,N],\hspace{1em}n=1,2,\ldots A sequence of positive fuzzy numbers (xn),n=1,2,…\left({x}_{n}),n=1,2,\ldots , is an unbounded if the norm ‖xn‖,n=1,2,…\Vert {x}_{n}\Vert ,n=1,2,\ldots , is an unbounded sequence.Definition 2.8xn{x}_{n}is said to be a positive solution of (5) if (xn)\left({x}_{n})is a sequence of positive fuzzy numbers, which satisfy (5). A positive fuzzy number xxis called a positive equilibrium of (5) if x=Ax1+x+Bx.x=\frac{Ax}{1+x+Bx}.Let (xn)\left({x}_{n})be a sequence of positive fuzzy numbers and xxis a positive fuzzy number, xn→x{x}_{n}\to xas n→∞n\to \infty if limn→∞D(xn,x)=0{\mathrm{lim}}_{n\to \infty }\hspace{0.33em}D\left({x}_{n},x)=0.3Main results3.1Existence of positive solution of population dynamics model (5)First, we study the existence of positive solutions of the population dynamics model (5). We need the following lemma.Lemma 3.1[30] Let f:R+×R+→R+f:{R}^{+}\times {R}^{+}\to {R}^{+}be continuous, and, A,BA,Bare fuzzy numbers. Then, (8)[f(A,B)]α=f([A]α,[B]α)α∈(0,1].{[f\left(A,B)]}^{\alpha }=f\left({\left[A]}^{\alpha },{\left[B]}^{\alpha })\hspace{1em}\alpha \in (0,1].Theorem 3.1If A,BA,B, and initial values x−2,x−1{x}_{-2},{x}_{-1}of population dynamics model (5) are positive fuzzy numbers, then, there exists a unique positive solution xn{x}_{n}of the population dynamics model (5).ProofSuppose that there exists a sequence of fuzzy numbers (xn)\left({x}_{n})satisfying (5) with initial condition x−1,x−2{x}_{-1},{x}_{-2}. Consider the α\alpha -cuts, α∈(0,1]\alpha \in (0,1], (9)[xn]α=[Ln,α,Rn,α],n=0,1,2,…{\left[{x}_{n}]}^{\alpha }=\left[{L}_{n,\alpha },{R}_{n,\alpha }],\hspace{1em}n=0,1,2,\ldots It follows from (5), (9), and Lemma 3.1 that [xn]α=[Ln,α,Rn,α]=Axn−11+xn−1+Bxn−2α=[A]α×[xn−1]α1+[xn−1]α+[B]α×[xn−2]α=[Al,αLn−1,α,Ar,αRn−1,α][1+Ln−1,α+Bl,αLn−2,α,1+Rn−1,α+Br,αRn−2,α].\begin{array}{rcl}{\left[{x}_{n}]}^{\alpha }& =& \left[{L}_{n,\alpha },{R}_{n,\alpha }]\\ & =& {\left[\frac{A{x}_{n-1}}{1+{x}_{n-1}+B{x}_{n-2}}\right]}^{\alpha }\\ & =& \frac{{\left[A]}^{\alpha }\times {\left[{x}_{n-1}]}^{\alpha }}{1+{\left[{x}_{n-1}]}^{\alpha }+{\left[B]}^{\alpha }\times {\left[{x}_{n-2}]}^{\alpha }}\\ & =& \frac{\left[{A}_{l,\alpha }{L}_{n-1,\alpha },{A}_{r,\alpha }{R}_{n-1,\alpha }]}{\left[1+{L}_{n-1,\alpha }+{B}_{l,\alpha }{L}_{n-2,\alpha },1+{R}_{n-1,\alpha }+{B}_{r,\alpha }{R}_{n-2,\alpha }]}.\end{array}Noting Remark 2.1, one of the following two cases holds:Case I: (10)[xn]α=[Ln,α,Rn,α]=Al,αLn−1,α1+Ln−1,α+Bl,αLn−2,α,Ar,αRn−1,α1+Rn−1,α+Br,αRn−2,α.{\left[{x}_{n}]}^{\alpha }=\left[{L}_{n,\alpha },{R}_{n,\alpha }]=\left[\frac{{A}_{l,\alpha }{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+{B}_{l,\alpha }{L}_{n-2,\alpha }},\frac{{A}_{r,\alpha }{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }+{B}_{r,\alpha }{R}_{n-2,\alpha }}\right].Case II: (11)[xn]α=[Ln,α,Rn,α]=Ar,αRn−1,α1+Rn−1,α+Br,αRn−2,α,Al,αLn−1,α1+Ln−1,α+Bl,αLn−2,α.{\left[{x}_{n}]}^{\alpha }=\left[{L}_{n,\alpha },{R}_{n,\alpha }]=\left[\frac{{A}_{r,\alpha }{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }+{B}_{r,\alpha }{R}_{n-2,\alpha }},\frac{{A}_{l,\alpha }{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+{B}_{l,\alpha }{L}_{n-2,\alpha }}\right].If Case I holds true, it follows that for n∈{0,1,2,…},α∈(0,1]n\in \left\{0,1,2,\ldots \right\},\alpha \in (0,1]: (12)Ln,α=Al,αLn−1,α1+Ln−1,α+Bl,αLn−2,α,Rn,α=Ar,αRn−1,α1+Rn−1,α+Br,αRn−2,α.{L}_{n,\alpha }=\frac{{A}_{l,\alpha }{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+{B}_{l,\alpha }{L}_{n-2,\alpha }},\hspace{1em}{R}_{n,\alpha }=\frac{{A}_{r,\alpha }{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }+{B}_{r,\alpha }{R}_{n-2,\alpha }}.□Next, the proof is similar to those of Theorem 3.1 [10]. We omit it.Remark 3.1From theoretical point of view, the existence of solution for fuzzy difference equation is very important with initial condition. Therefore, in this sense, the existence of positive fuzzy solution for discrete population dynamics model is of vital importance and practical significance. In fact, according to Theorem 3.1, the positive solution of discrete population dynamics model with fuzzy state is a sequence of positive fuzzy numbers, which can describe the fuzzy uncertainty of the dynamics model.3.2Dynamics of discrete population model (5)To study the dynamical behavior of the positive solutions of the discrete population model (5), according to Definition 2.6, we consider two cases.First, if Case I holds true, the following lemma is essential to the proof of next theorem.Lemma 3.2Consider the difference equation: (13)yn=kyn−11+yn−1+βyn−2,n=1,2,…,{y}_{n}=\frac{k{y}_{n-1}}{1+{y}_{n-1}+\beta {y}_{n-2}},\hspace{1em}n=1,2,\ldots ,where y−2,y−1∈(0,+∞){y}_{-2},{y}_{-1}\in \left(0,+\infty ), if k>1,0≤β<1k\gt 1,0\le \beta \lt 1, then the following statements are true. (i)The system exists a trivial equilibrium point y∗=0{y}^{\ast }=0, which is unstable.(ii)The system exists a unique positive equilibrium y∗=k−1β+1{y}^{\ast }=\frac{k-1}{\beta +1}, which is globally asymptotically stable.(iii)Every positive solution yn{y}_{n}of (13) is bounded and persistent.Proof (i) Let y∗{y}^{\ast }be an equilibrium point of (13). It is easy to obtain that there exist two equilibrium points: y∗=0,andy∗=k−1β+1.{y}^{\ast }=0,\hspace{1.0em}{\rm{and}}\hspace{1.0em}{y}^{\ast }=\frac{k-1}{\beta +1}.It is clear that the trivial equilibrium point y∗{y}^{\ast }is unstable. So we omit it.(ii) The linearized equation associated with (13) at equilibrium expressed as follows: y∗=k−1β+1{y}^{\ast }=\frac{k-1}{\beta +1}is expressed as follows: yn−kβ+1k(β+1)yn−1+β(k−1)k(β+1)yn−2=0.{y}_{n}-\frac{k\beta +1}{k\left(\beta +1)}{y}_{n-1}+\frac{\beta \left(k-1)}{k\left(\beta +1)}{y}_{n-2}=0.\hspace{1.8em}Since k>1,0≤β<1k\gt 1,0\le \beta \lt 1, it is easy to obtain (14)kβ+1k(β+1)+β(k−1)k(β+1)<1.\frac{k\beta +1}{k\left(\beta +1)}+\frac{\beta \left(k-1)}{k\left(\beta +1)}\lt 1.By virtue of Theorem 1.3.7 in [7], we have that the system is locally asymptotically stable.On the other hand, it is similar to the proof of Theorem 1 in [32], and we can show that limn→∞yn=k−1β+1{\mathrm{lim}}_{n\to \infty }\hspace{0.33em}{y}_{n}=\frac{k-1}{\beta +1}. Thus, the unique positive equilibrium y∗=k−1β+1{y}^{\ast }=\frac{k-1}{\beta +1}is globally asymptotically stable.(iii) Let yn{y}_{n}be a solution of (13). We consider the following difference equation: (15)un=kun−11+un−1,n=1,2,3,…,{u}_{n}=\frac{k{u}_{n-1}}{1+{u}_{n-1}},\hspace{1em}n=1,2,3,\ldots ,and the initial values of (15) are satisfied with (16)y0≤u0.{y}_{0}\le {u}_{0}.It follows from (13), (15), and (16) that (17)yn=kyn−11+yn−1+βyn−2≤kyn−11+yn−1≤kun−11+un−1=un,n≥2.{y}_{n}=\frac{k{y}_{n-1}}{1+{y}_{n-1}+\beta {y}_{n-2}}\le \frac{k{y}_{n-1}}{1+{y}_{n-1}}\le \frac{k{u}_{n-1}}{1+{u}_{n-1}}={u}_{n},\hspace{1em}n\ge 2.It is clear that every solution un{u}_{n}of (15) converges to equilibrium k−1k-1, i.e., limn→∞un=k−1{\mathrm{lim}}_{n\to \infty }\hspace{0.25em}{u}_{n}=k-1. Therefore, it follows from (17) that (18)yn≤limn→∞supyn≤limn→∞supun=limn→∞un=k−1.\hspace{-21.15em}{y}_{n}\le \mathop{\mathrm{lim}}\limits_{n\to \infty }\sup {y}_{n}\le \mathop{\mathrm{lim}}\limits_{n\to \infty }\sup {u}_{n}=\mathop{\mathrm{lim}}\limits_{n\to \infty }{u}_{n}=k-1.On the other hand, we consider the family of sequences {un(ε)}\left\{{u}_{n}\left(\varepsilon )\right\}, where (19)un(ε)=kun−1(ε)1+un−1(ε)+β(k−1+ε),n=2,3,….{u}_{n}\left(\varepsilon )=\frac{k{u}_{n-1}\left(\varepsilon )}{1+{u}_{n-1}\left(\varepsilon )+\beta \left(k-1+\varepsilon )},\hspace{1em}n=2,3,\ldots .Without loss of generality, we take 0<ε<(k−1)(1−β)β0\lt \varepsilon \lt \frac{\left(k-1)\left(1-\beta )}{\beta }. For every fixed ε\varepsilon , difference equation (19) has a stationary trajectory such that u(ε)=limn→∞un(ε)=(k−1)(1−β)−βεu\left(\varepsilon )={\mathrm{lim}}_{n\to \infty }\hspace{0.33em}{u}_{n}\left(\varepsilon )=\left(k-1)\left(1-\beta )-\beta \varepsilon . From (ii), there exists an n0∈N{n}_{0}\in Nsuch that yn≤k−1+ε{y}_{n}\le k-1+\varepsilon for n≥n0n\ge {n}_{0}.Let the initial conditions of (19) be positive and satisfy the conditions: (20)un0+1(ε)≤yn0+1.{u}_{{n}_{0}+1}\left(\varepsilon )\le {y}_{{n}_{0}+1}.Then, we obtain that (21)yn=kyn−11+yn−1+βyn−2≥kyn−11+yn−1+β(k−1+ε)≥kun−1(ε)1+un−1(ε)+β(k−1+ε)=un(ε).{y}_{n}=\frac{k{y}_{n-1}}{1+{y}_{n-1}+\beta {y}_{n-2}}\ge \frac{k{y}_{n-1}}{1+{y}_{n-1}+\beta \left(k-1+\varepsilon )}\ge \frac{k{u}_{n-1}\left(\varepsilon )}{1+{u}_{n-1}\left(\varepsilon )+\beta \left(k-1+\varepsilon )}={u}_{n}\left(\varepsilon ).Therefore, yn≥un(ε){y}_{n}\ge {u}_{n}\left(\varepsilon ), limn→∞infyn≥limn→∞infun(ε)=limn→∞un(ε)=(k−1)(1−β){\mathrm{lim}}_{n\to \infty }\inf {y}_{n}\ge {\mathrm{lim}}_{n\to \infty }\inf {u}_{n}\left(\varepsilon )={\mathrm{lim}}_{n\to \infty }{u}_{n}\left(\varepsilon )=\left(k-1)\left(1-\beta ).So yn≥(k−1)(1−β){y}_{n}\ge \left(k-1)\left(1-\beta )for n≥n0n\ge {n}_{0}. This completes the proof.□Theorem 3.2Consider the discrete population model (5), where parameters AAand BBand the initial conditions x−1,x0{x}_{-1},{x}_{0}are positive fuzzy numbers. If(22)Al,α>1,Br,α<1,α∈(0,1],{A}_{l,\alpha }\gt 1,\hspace{0.33em}{B}_{r,\alpha }\lt 1,\hspace{0.33em}\alpha \in (0,1],then the following statements are true. (i)Every positive solution xn{x}_{n}of discrete fuzzy population model (5) is bounded and persistent.(ii)Every positive solution xn{x}_{n}of discrete fuzzy population model (5) tends to the positive equilibrium point xxas n→∞n\to \infty .Proof (i) Since A,BA,Band the initial value x−1,x0{x}_{-1},{x}_{0}are positive fuzzy numbers, there exist positive real numbers MA,NA,MB,NB,M0,N0,M−1,N−1{M}_{A},{N}_{A},{M}_{B},{N}_{B},{M}_{0},{N}_{0},{M}_{-1},{N}_{-1}such that, for all α∈(0,1]\alpha \in (0,1], (23)[Al,α,Ar,α]⊂[MA,NA],[Bl,α,Br,α]⊂[MB,NB],[L0,α,R0,α]⊂[M0,N0],[L−l,α,R−1,α]⊂[M−1,N−1].\left[{A}_{l,\alpha },{A}_{r,\alpha }]\subset \left[{M}_{A},{N}_{A}],\hspace{1em}\left[{B}_{l,\alpha },{B}_{r,\alpha }]\subset \left[{M}_{B},{N}_{B}],\hspace{1em}\left[{L}_{0,\alpha },{R}_{0,\alpha }]\subset \left[{M}_{0},{N}_{0}],\hspace{1em}\left[{L}_{-l,\alpha },{R}_{-1,\alpha }]\subset \left[{M}_{-1},{N}_{-1}].Let xn{x}_{n}be a positive solution of discrete fuzzy population model (5), from (22), (23), and Lemma 3.2, we obtain (24)Ln,α>(Al,α−1)(1−Bl,α)>(MA−1)(1−NB)≔M,Rn,α<Ar,α−1<NA−1≔N{L}_{n,\alpha }\gt \left({A}_{l,\alpha }-1)\left(1-{B}_{l,\alpha })\gt \left({M}_{A}-1)\left(1-{N}_{B}):= M,\hspace{1em}{R}_{n,\alpha }\lt {A}_{r,\alpha }-1\lt {N}_{A}-1:= NFrom which, we get for n≥2,⋃α∈(0,1][Ln,α,Rn,α]⊂[M,N]n\ge 2,{\bigcup }_{\alpha \in (0,1]}\left[{L}_{n,\alpha },{R}_{n,\alpha }]\subset \left[M,N], and so ⋃α∈(0,1][Ln,α,Rn,α]¯⊆[M,N]\overline{{\bigcup }_{\alpha \in (0,1]}\left[{L}_{n,\alpha },{R}_{n,\alpha }]}\subseteq \left[M,N]. Thus, the positive solution is bounded and persistent.(ii) Suppose that there exists a fuzzy number xxsuch that (25)x=Ax1+x+Bx,[x]α=[Lα,Rα],α∈(0,1].x=\frac{Ax}{1+x+Bx},\hspace{1em}{\left[x]}^{\alpha }=\left[{L}_{\alpha },{R}_{\alpha }],\hspace{1em}\alpha \in (0,1].where Lα,Rα≥0{L}_{\alpha },{R}_{\alpha }\ge 0. Then, from (25), we can prove that (26)Lα=Al,αLα1+Lα+Bl,αLα,Rα=Ar,αRα1+Rα+Br,αRα.{L}_{\alpha }=\frac{{A}_{l,\alpha }{L}_{\alpha }}{1+{L}_{\alpha }+{B}_{l,\alpha }{L}_{\alpha }},\hspace{1em}{R}_{\alpha }=\frac{{A}_{r,\alpha }{R}_{\alpha }}{1+{R}_{\alpha }+{B}_{r,\alpha }{R}_{\alpha }}.\hspace{1.15em}Hence, from (26), we have (27)Lα=Al,α−1Bl,α+1,Rα=Ar,α−1Br,α+1.{L}_{\alpha }=\frac{{A}_{l,\alpha }-1}{{B}_{l,\alpha }+1},\hspace{1em}{R}_{\alpha }=\frac{{A}_{r,\alpha }-1}{{B}_{r,\alpha }+1}.Let xn{x}_{n}be a positive solution of discrete fuzzy population model (5). Since (22) holds true, we can apply Lemma 3.2 to system (13), and so we have (28)limn→∞Ln,α=Lα,limn→∞Rn,α=Rα.\mathop{\mathrm{lim}}\limits_{n\to \infty }{L}_{n,\alpha }={L}_{\alpha },\hspace{1em}\mathop{\mathrm{lim}}\limits_{n\to \infty }{R}_{n,\alpha }={R}_{\alpha }.Therefore, from (28), we have limn→∞D(xn,x)=limn→∞supα∈(0,1]{max{∣Ln,α−Lα∣,∣Rn,α−Rα∣}}=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }D\left({x}_{n},x)=\mathop{\mathrm{lim}}\limits_{n\to \infty }\mathop{\sup }\limits_{\alpha \in (0,1]}\left\{{\rm{\max }}\left\{| {L}_{n,\alpha }-{L}_{\alpha }| ,| {R}_{n,\alpha }-{R}_{\alpha }| \right\}\right\}=0.This completes the proof of the theorem.□Second, if Case II holds true, it follows that for n∈{0,1,2,…},α∈(0,1]n\in \left\{0,1,2,\ldots \right\},\alpha \in (0,1](29)Ln,α=Ar,αRn−1,α1+Rn−1,α+Br,αRn−2,α,Rn,α=Al,αLn−1,α1+Ln−1,α+Bl,αRn−2,α.{L}_{n,\alpha }=\frac{{A}_{r,\alpha }{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }+{B}_{r,\alpha }{R}_{n-2,\alpha }},\hspace{1em}{R}_{n,\alpha }=\frac{{A}_{l,\alpha }{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+{B}_{l,\alpha }{R}_{n-2,\alpha }}.We need the following lemmas.Lemma 3.3Consider the system of difference equations: (30)yn=azn−11+zn−1+czn−2,zn=ayn−11+yn−1+cyn−2,n=1,2,…,{y}_{n}=\frac{a{z}_{n-1}}{1+{z}_{n-1}+c{z}_{n-2}},\hspace{1em}{z}_{n}=\frac{a{y}_{n-1}}{1+{y}_{n-1}+c{y}_{n-2}},\hspace{1em}n=1,2,\ldots ,where a,c,y−1,y0,z−1,z0∈(0,+∞)a,c,{y}_{-1},{y}_{0},{z}_{-1},{z}_{0}\in \left(0,+\infty ). If(31)a>1,a\gt 1,then the following statements are true.(i)The system exists a trivial equilibrium point (0,0)\left(0,0), which is unstable.(ii)The system exists a unique positive equilibrium point (y∗,z∗)=a−1c+1,a−1c+1({y}^{\ast },{z}^{\ast })=\left(\frac{a-1}{c+1},\frac{a-1}{c+1}\right), which is locally asymptotically stable.Proof(i) The linearized equation of system (30) about (0,0)\left(0,0)is expressed as follows: (32)Ψn=GΨn−1,{\Psi }_{n}=G{\Psi }_{n-1},where Ψn−1=yn−1yn−2zn−1zn−2,G=00a01000a0000010.{\Psi }_{n-1}=\left(\begin{array}{c}{y}_{n-1}\\ {y}_{n-2}\\ {z}_{n-1}\\ {z}_{n-2}\end{array}\right),\hspace{1em}G=\left(\begin{array}{cccc}0& 0& a& 0\\ 1& 0& 0& 0\\ a& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right).The characteristic equation of (32) is expressed as follows: (33)λ2(λ2−a2)=0.{\lambda }^{2}\left({\lambda }^{2}-{a}^{2})=0.It follows from (31) that there are two roots of characteristic equation outside the unit disk. So the trivial equilibrium (0,0)\left(0,0)is unstable.(ii) The linearized equation of system (30) about (y∗,z∗)({y}^{\ast },{z}^{\ast })is expressed as follows: (34)Ψn=HΨn−1,{\Psi }_{n}=H{\Psi }_{n-1},where H=00a+acz∗(1+z∗+cz∗)2−acz∗(1+z∗+cz∗)21000a+acy∗(1+y∗+cy∗)2−acy∗(1+y∗+cy∗)2000010.H=\left(\begin{array}{cccc}0& 0& \frac{a+ac{z}^{\ast }}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}& -\frac{ac{z}^{\ast }}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}\\ 1& 0& 0& 0\\ \frac{a+ac{y}^{\ast }}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}& -\frac{ac{y}^{\ast }}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}& 0& 0\\ 0& 0& 1& 0\end{array}\right).Let λ1,λ2,λ3,λ4{\lambda }_{1},{\lambda }_{2},{\lambda }_{3},{\lambda }_{4}denote the eigenvalues of matrix HH, let K=diag(m1,m2,m3,m4)K=\hspace{0.1em}\text{diag}\hspace{0.1em}\left({m}_{1},{m}_{2},{m}_{3},{m}_{4})be a diagonal matrix, where m1=m3=1,mk=1−kε(k=2,4){m}_{1}={m}_{3}=1,{m}_{k}=1-k\varepsilon \left(k=2,4), and (35)0<ε<141−acz∗(1+z∗+cz∗)2−a−acz∗.0\lt \varepsilon \lt \frac{1}{4}\left(1-\frac{ac{z}^{\ast }}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}-a-ac{z}^{\ast }}\right).Clearly, KKis invertible. Computing matrix KHK−1KH{K}^{-1}, we obtain that (36)KHK−1=00(a+acz∗)m1m3−1(1+z∗+cz∗)2−acz∗m1m4−1(1+z∗+cz∗)2m2m1−1000(a+acy∗)m3m1−1(1+y∗+cy∗)2−acy∗m3m2−1(1+y∗+cy∗)20000m4m3−10.KH{K}^{-1}=\left(\begin{array}{cccc}0& 0& \frac{\left(a+ac{z}^{\ast }){m}_{1}{m}_{3}^{-1}}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}& -\frac{ac{z}^{\ast }{m}_{1}{m}_{4}^{-1}}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}\\ {m}_{2}{m}_{1}^{-1}& 0& 0& 0\\ \frac{\left(a+ac{y}^{\ast }){m}_{3}{m}_{1}^{-1}}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}& -\frac{ac{y}^{\ast }{m}_{3}{m}_{2}^{-1}}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}& 0& 0\\ 0& 0& {m}_{4}{m}_{3}^{-1}& 0\end{array}\right).It is obtained from (36) that a+acz∗(1+z∗+cz∗)2+acz∗m4−1(1+z∗+cz∗)2=a+acz∗1−11−4ε(1+z∗+cz∗)2<1\frac{a+ac{z}^{\ast }}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}+\frac{ac{z}^{\ast }{m}_{4}^{-1}}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}=\frac{a+ac{z}^{\ast }\left(1-\frac{1}{1-4\varepsilon }\right)}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}}\lt 1a+acy∗(1+y∗+cy∗)2+acy∗m2−1(1+y∗+cy∗)2=a+acy∗1−11−2ε(1+y∗+cy∗)2<1.\frac{a+ac{y}^{\ast }}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}+\frac{ac{y}^{\ast }{m}_{2}^{-1}}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}=\frac{a+ac{y}^{\ast }\left(1-\frac{1}{1-2\varepsilon }\right)}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}\lt 1.It is well known that HHhas the same eigenvalues as KHK−1KH{K}^{-1}, and we have (37)max1≤i≤4∣λi∣≤‖KHK−1‖∞=maxm2m1−1,m4m3−1,a+acz∗+acz∗m1m4−1(1+z∗+cz∗)2,a+acy∗+acy∗m3m2−1(1+y∗+cy∗)2<1.\mathop{\max }\limits_{1\le i\le 4}| {\lambda }_{i}| \le \Vert KH{K}^{-1}{\Vert }_{\infty }=\max \left\{{m}_{2}{m}_{1}^{-1},{m}_{4}{m}_{3}^{-1},\frac{a+ac{z}^{\ast }+ac{z}^{\ast }{m}_{1}{m}_{4}^{-1}}{{\left(1+{z}^{\ast }+c{z}^{\ast })}^{2}},\frac{a+ac{y}^{\ast }+ac{y}^{\ast }{m}_{3}{m}_{2}^{-1}}{{\left(1+{y}^{\ast }+c{y}^{\ast })}^{2}}\right\}\lt 1.This implies that the equilibrium (y∗,z∗)({y}^{\ast },{z}^{\ast })of (30) is locally asymptotically stable.□Theorem 3.3Suppose that parameters A and B of discrete fuzzy population model (5) are positive trivial fuzzy numbers (positive real numbers) and A>1,0<B<1A\gt 1,0\lt B\lt 1. If the following condition are satisfied, (38)1+BLn−2,α1+BRn−2,α≤Ln−1,αRn−1,α,n=1,2,…,∀α∈(0,1].\frac{1+B{L}_{n-2,\alpha }}{1+B{R}_{n-2,\alpha }}\le \frac{{L}_{n-1,\alpha }}{{R}_{n-1,\alpha }},\hspace{1em}n=1,2,\ldots ,\forall \alpha \in (0,1].Then, the following statements are true. (i)Every positive solution xn{x}_{n}of discrete fuzzy population model (5) is bounded and persistence.(ii)Every positive solution xn{x}_{n}of discrete fuzzy population model (5) tends to the positive equilibrium point xxas n→+∞n\to +\infty .Proof (i) The proof is similar to those of (i) in Theorem 3.2. Let xn{x}_{n}be a positive solution of discrete fuzzy population model (5). It is easy to get that Ln,α≥(A−1)(1−B),Rn,α≤A−1,∀α∈(0,1].{L}_{n,\alpha }\ge \left(A-1)\left(1-B),\hspace{1em}{R}_{n,\alpha }\le A-1,\hspace{1em}\forall \alpha \in (0,1].From which, we obtain for n≥2,⋃α∈(0,1][Ln,α,Rn,α]⊂[(A−1)(1−B),A−1]n\ge 2,{\bigcup }_{\alpha \in (0,1]}\left[{L}_{n,\alpha },{R}_{n,\alpha }]\subset \left[\left(A-1)\left(1-B),A-1], and so ⋃α∈(0,1][Ln,α,Rn,α]¯⊆[(A−1)(1−B),A−1]\overline{{\bigcup }_{\alpha \in (0,1]}\left[{L}_{n,\alpha },{R}_{n,\alpha }]}\hspace{0.25em}\subseteq \left[\left(A-1)\left(1-B),A-1]. Thus, the positive solution is bounded and persistent.(ii) Suppose that there exists a positive fuzzy number xxsatisfying (25). Then from (25), we have (39)Lα=ARα1+Rα+BRα,Rα=ALα1+Lα+BLα,∀α∈(0,1].{L}_{\alpha }=\frac{A{R}_{\alpha }}{1+{R}_{\alpha }+B{R}_{\alpha }},\hspace{1em}{R}_{\alpha }=\frac{A{L}_{\alpha }}{1+{L}_{\alpha }+B{L}_{\alpha }},\forall \alpha \in (0,1].From (39), we have Lα=Rα=A−1B+1.{L}_{\alpha }={R}_{\alpha }=\frac{A-1}{B+1}.Let xn{x}_{n}be a positive solution of discrete fuzzy population model (5) such that Case II holds. Namely, (40)Ln,α=ARn−1,α1+Rn−1,α++BRn−2,α,Rn,α=ALn−1,α1+Ln−1,α+BLn−2,α.{L}_{n,\alpha }=\frac{A{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }++B{R}_{n-2,\alpha }},\hspace{1em}{R}_{n,\alpha }=\frac{A{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+B{L}_{n-2,\alpha }}.Since A>1,0<B<1A\gt 1,0\lt B\lt 1and (38) is satisfied, we can apply Lemma 3.3 to system (40), and so we have (41)limn→∞Ln,α=Lα,limn→∞Rn,α=Rα.\mathop{\mathrm{lim}}\limits_{n\to \infty }{L}_{n,\alpha }={L}_{\alpha },\hspace{1em}\mathop{\mathrm{lim}}\limits_{n\to \infty }{R}_{n,\alpha }={R}_{\alpha }.Therefore, from (41), we have limn→∞D(xn,x)=limn→∞supα∈(0,1]{max{∣Ln,α−Lα∣,∣Rn,α−Rα∣}}=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }D\left({x}_{n},x)=\mathop{\mathrm{lim}}\limits_{n\to \infty }\mathop{\sup }\limits_{\alpha \in (0,1]}\left\{{\rm{\max }}\left\{| {L}_{n,\alpha }-{L}_{\alpha }| ,| {R}_{n,\alpha }-{R}_{\alpha }| \right\}\right\}=0.This completes the proof of Theorem 3.3.□Remark 3.2In the population dynamical model, the parameters of model are derived from statistic data with vagueness or uncertainty. It corresponds to reality to use fuzzy parameters in the population dynamical model. In contrast with the classic population model, the solution of fuzzy population model is within a range of value (approximate value), which are taken into account fuzzy uncertainties. Furthermore, the global asymptotic behavior of the discrete second-order population model are obtained in fuzzy context.4Numerical examplesExample 4.1Consider the following second-order fuzzy discrete population model: (42)xn=Axn−11+xn−1+Bxn−2,n=1,2,…,{x}_{n}=\frac{A{x}_{n-1}}{1+{x}_{n-1}+B{x}_{n-2}},\hspace{1em}n=1,2,\ldots ,and we take A,B,A,B,and the initial values x−1,x0{x}_{-1},{x}_{0}such that (43)A(x)=2x−3,1.5≤x≤2−2x+5,2≤x≤2.5,x−1(x)=2x−1,0.5≤x≤1−5x+6,1≤x≤1.2\begin{array}{ll}A\left(x)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}2x-3,& 1.5\le x\le 2\\ -2x+5,& 2\le x\le 2.5\end{array}\right.,& {x}_{-1}\left(x)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}2x-1,& 0.5\le x\le 1\\ -5x+6,& 1\le x\le 1.2\end{array}\right.\end{array}(44)B(x)=5x−2,0.4≤x≤0.6−5x+4,0.6≤x≤0.8,x0(x)=x−3,3≤x≤4−x+5,4≤x≤5.\begin{array}{ll}B\left(x)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}5x-2,& 0.4\le x\le 0.6\\ -5x+4,& 0.6\le x\le 0.8\end{array}\right.,& {x}_{0}\left(x)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}x-3,& 3\le x\le 4\\ -x+5,& 4\le x\le 5.\end{array}\right.\end{array}From (43), we obtain (45)[A]α=1.5+12α,2.5−12α,[x−1]α=0.5+12α,1.2−15α,α∈(0,1].{\left[A]}^{\alpha }=\left[1.5+\frac{1}{2}\alpha ,2.5-\frac{1}{2}\alpha \right],\hspace{1em}{\left[{x}_{-1}]}^{\alpha }=\left[0.5+\frac{1}{2}\alpha ,1.2-\frac{1}{5}\alpha \right],\hspace{1em}\alpha \in (0,1].From (44), we obtain (46)[B]α=0.4+15α,0.8−15α,[x0]α=[3+α,5−α],α∈(0,1].{\left[B]}^{\alpha }=\left[0.4+\frac{1}{5}\alpha ,0.8-\frac{1}{5}\alpha \right],\hspace{1em}{\left[{x}_{0}]}^{\alpha }={[}3+\alpha ,5-\alpha ],\hspace{1em}\alpha \in (0,1].Therefore, it follows that (47)⋃α∈(0,1][A]α¯=[1.5,2.5],⋃α∈(0,1][x−1]α¯=[0.5,1.2],⋃α∈(0,1][B]α¯=[0.4,0.8],⋃α∈(0,1][x0]α¯=[3,5].\overline{\bigcup _{\alpha \in (0,1]}{\left[A]}^{\alpha }}=\left[1.5,2.5],\hspace{1em}\overline{\bigcup _{\alpha \in (0,1]}{\left[{x}_{-1}]}^{\alpha }}=\left[0.5,1.2],\hspace{1em}\overline{\bigcup _{\alpha \in (0,1]}{\left[B]}^{\alpha }}=\left[0.4,0.8],\hspace{1em}\overline{\bigcup _{\alpha \in (0,1]}{\left[{x}_{0}]}^{\alpha }}=\left[3,5].From (42), it results in a coupled system of difference equations with parameter α\alpha , (48)Ln,α=Al,αLn−1,α1+Ln−1,α+Bl,αLn−1,α,Rn,α=Ar,αRn−1,α1+Rn−1,α+Br,αRn−1,α,α∈(0,1].{L}_{n,\alpha }=\frac{{A}_{l,\alpha }{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+{B}_{l,\alpha }{L}_{n-1,\alpha }},\hspace{1em}{R}_{n,\alpha }=\frac{{A}_{r,\alpha }{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }+{B}_{r,\alpha }{R}_{n-1,\alpha }},\hspace{1em}\alpha \in (0,1].Therefore, Al,α>1,Br,α<1,∀α∈(0,1]{A}_{l,\alpha }\gt 1,{B}_{r,\alpha }\lt 1,\forall \alpha \in (0,1], and initial values x0{x}_{0}are positive fuzzy numbers, so from Theorem 3.2, we have that every positive solution xn{x}_{n}of equation (42) is bounded and persistence. In addition, from Theorem 3.2, equation (42) has a unique positive equilibrium x¯=(0.357,0.625,0.833)\overline{x}=\left(0.357,0.625,0.833). Moreover, every positive solution xn{x}_{n}of equation (42) converges the unique equilibrium x¯\overline{x}with respect to DDas n→∞n\to \infty (see Figures 1, 2, 3).Figure 1The dynamics of system (42).Figure 2The solution of system (48) at α=0\alpha =0and α=0.25\alpha =0.25.Figure 3The solution of system (48) at α=0.75\alpha =0.75and α=1\alpha =1.Example 4.2Consider the second-order fuzzy discrete population model (42), where A=1.5,B=0.6A=1.5,B=0.6, and the initial values x0,x−1{x}_{0},{x}_{-1}are satisfied (49)x−1(x)=0.5x−0.5,1≤x≤3−0.5x+2.5,3≤x≤5,x0(x)=2x−6,3≤x≤3.5−2x+8,3.5≤x≤4.{x}_{-1}\left(x)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}0.5x-0.5,& 1\le x\le 3\\ -0.5x+2.5,& 3\le x\le 5\end{array}\right.\hspace{0.33em},\hspace{1em}{x}_{0}\left(x)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}2x-6,& 3\le x\le 3.5\\ -2x+8,& 3.5\le x\le 4.\end{array}\right.From (48), we obtain (50)[x−1]α=[1+2α,5−2α],[x0]α=3+12α,4−12α,α∈(0,1].{\left[{x}_{-1}]}^{\alpha }={[}1+2\alpha ,5-2\alpha ],\hspace{1em}{\left[{x}_{0}]}^{\alpha }=\left[3+\frac{1}{2}\alpha ,4-\frac{1}{2}\alpha \right],\hspace{1em}\alpha \in (0,1].Therefore, it follows that (51)⋃α∈(0,1][x−1]α¯=[1,5],⋃α∈(0,1][x0]α¯=[3,4].\overline{\bigcup _{\alpha \in (0,1]}{\left[{x}_{-1}]}^{\alpha }}=\left[1,5],\hspace{1em}\overline{\bigcup _{\alpha \in (0,1]}{\left[{x}_{0}]}^{\alpha }}=\left[3,4].From (42), it results in a coupled system of difference equation with parameter α\alpha , (52)Ln,α=ARn−1,α1+Rn−1,α+BRn−2,α,Rn,α=ALn−1,α1+Ln−1,α+BLn−2,α,α∈(0,1].{L}_{n,\alpha }=\frac{A{R}_{n-1,\alpha }}{1+{R}_{n-1,\alpha }+B{R}_{n-2,\alpha }},\hspace{1em}{R}_{n,\alpha }=\frac{A{L}_{n-1,\alpha }}{1+{L}_{n-1,\alpha }+B{L}_{n-2,\alpha }},\hspace{1em}\alpha \in (0,1].It is clear that (38) is satisfied and initial values x−1,x0{x}_{-1},{x}_{0}are positive fuzzy numbers, so from Theorem 3.3, equation (42) has a unique positive equilibrium x¯=0.3125\overline{x}=0.3125. Moreover, every positive solution xn{x}_{n}of equation (42) converges the unique equilibrium x¯\overline{x}with respect to DDas n→∞n\to \infty (see Figures 4, 5, 6).Figure 4The dynamics of system (42).Figure 5The solution of system (48) at α=0\alpha =0and α=0.25\alpha =0.25.Figure 6The solution of system (48) at α=0.75\alpha =0.75and α=1\alpha =1.5ConclusionIn this work, according to a generalization of division (gg-division) of fuzzy number, we study the second-order fuzzy discrete population model xn=Axn−11+xn−1+Bxn−2{x}_{n}=\frac{A{x}_{n-1}}{1+{x}_{n-1}+B{x}_{n-2}}. The existence of the positive solution and the qualitative behavior to (5) are investigated. The main results are as follows: (1)Under Case I, the positive solution is bounded and persistent if Al,α>1,Br,α<1,α∈(0,1]{A}_{l,\alpha }\gt 1,{B}_{r,\alpha }\lt 1,\alpha \in (0,1]. Every positive solution xn{x}_{n}tends to the unique equilibrium xxas n→∞n\to \infty .(2)Under Case II, the positive solution is bounded and persistent if A,BA,Bare positive trivial fuzzy numbers and A>1,B<1,1+BLn−2,α1+BRn−2,α≤Ln−1,αRn−1,α,n=1,2,…,∀α∈(0,1]A\gt 1,B\lt 1,\frac{1+B{L}_{n-2,\alpha }}{1+B{R}_{n-2,\alpha }}\le \frac{{L}_{n-1,\alpha }}{{R}_{n-1,\alpha }},\hspace{0.33em}n=1,2,\ldots ,\forall \alpha \in (0,1]. Every positive solution xn{x}_{n}tends to the unique equilibrium xxas n→∞n\to \infty .

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: fuzzy discrete population model; g -division; boundedness; global asymptotic behavior; 39A10

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