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Global Behavior for a Diffusive Predator-Prey Model with Stage Structure and Nonlinear Density Restriction-II: The Case in ℝ 1

Global Behavior for a Diffusive Predator-Prey Model with Stage Structure and Nonlinear Density... Global Behavior for a Diffusive Predator-Prey Model with Stage Structure and Nonlinear Density Restriction-II: The Case in ℝ1 <meta name="citation_title" content="Global Behavior for a Diffusive Predator-Prey Model with Stage Structure and Nonlinear Density Restriction-II: The Case in ℝ 1 " /> //// Hindawi Publishing Corporation Home Journals About Us About this Journal Submit a Manuscript Table of Contents Journal Menu Abstracting and Indexing Aims and Scope Article Processing Charges Articles in Press Author Guidelines Bibliographic Information Contact Information Editorial Board Editorial Workflow Free eTOC Alerts Reviewers Acknowledgment Subscription Information Open Special Issues Published Special Issues Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Linked References How to Cite this Article Boundary Value Problems Volume 2009 (2009), Article ID 654539, 19 pages doi:10.1155/2009/654539 Research Article Global Behavior for a Diffusive Predator-Prey Model with Stage Structure and Nonlinear Density Restriction-II: The Case in ℝ 1 Rui Zhang , 1,2 Ling Guo , 1 and Shengmao Fu 1 1 Department of Mathematics, Northwest Normal University, Lanzhou 730070, China 2 Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China Received 2 April 2009; Accepted 31 August 2009 Academic Editor: Wenming Zou Copyright © 2009 Rui Zhang et al. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. <h4>Abstract</h4> A Holling type III predator-prey model with self- and cross-population pressure is considered. Using the energy estimate and Gagliardo-Nirenberg-type inequalities, the existence and uniform boundedness of global solutions to the model are dicussed. In addition, global asymptotic stability of the positive equilibrium point for the model is proved by Lyapunov function. 1. Introduction This paper is a continuation of Part I [ 1 ]. In Section of Part I, using the energy estimate and bootstrap arguments, the global existence of solutions for a Holling type III cross-diffusion predator-prey model with stage-structure has been discussed when the space dimension be less than 6. However, to obtain the estimate for the population density of predator species, there is not cross-diffusion for in Part I. All diffusive predator-prey systems behave, more or less, in the same way, for both semilinear and cross-diffusive models, at least for small values of the cross diffusivities. Consequently, all the available information for linear diffusive models is essential to realize the behavior of the most complicated cross-diffusive systems [ 2 – 17 ]. In this paper, we consider the following cross-diffusion system: (1.1) where and are positive constants. Also, are linear diffusion coefficients of , respectively, while are referred as self-diffusion pressures, and are cross-diffusion pressures. If , then ( 1.1 ) reduces to the system (1.4) of Part I. Recently, the work in [ 18 – 20 ] studied the existence, uniform boundedness, and uniform convergence of global solutions for the Lotka-Volterra cross-diffusion models without stage-structure in the case that the space dimension . In this paper, we consider mainly the existence and uniform boundedness of global solutions for the model ( 1.1 ) with nonlinear density restriction and stage-structure. Moreover, global asymptotic stability of the positive equilibrium point for ( 1.1 ) is proved by an important lemma of [ 21 ]. The proof is complete and complement the uniform convergence theorem in [ 18 – 20 ]. 2. Global Existence and Uniform Boundedness For simplicity, denote . The local existence result of solutions to ( 1.1 ) is an immediate consequence of a series of papers [ 22 , 23 ] by Amann. Roughly speaking, if , then ( 1.1 ) has a unique nonnegative solution , where is the maximal existence time for the solution. If satisfies (2.1) then . If, in addition, , then . The main result in this section is as follows. Theorem 2.1. Let is the unique nonnegative solution of ( 1.1 ) in its maximal existence interval . Assume that (2.2) Then there exists and positive constants which depend on , such that (2.3) (2.4) and . In particular, if , where and are positive constants, then depend on , but do not depend on . The following Gagliardo-Nirenberg-type inequalities and corresponding corollary play an importance role in the proof of Theorem 2.1 . Theorem 2.2 (see [ 18 ]). Let be a bounded domain with . For every function , the derivative satisfies the inequality (2.5) provided one of the following three conditions is satisfied: , or , and is not a nonnegative integer, where , for all , and the positive constant depends on . Corollary 2.3. There exists a positive constant such that (2.6) (2.7) (2.8) (2.9) For simplicity, denote that is Sobolev embedding constant or other kind of absolute constant. are some positive constants which depend on . Also, are positive constants which depend on . When do not depend on , but on . Proof of Theorem 2.1 . Step 1. Estimate , . Firstly, taking integration of the first and second equations in ( 2.7 ) over the domain , respectively, and combining the two integration equalities linearly, we have (2.10) From Young inequality and Hölder inequality, we can see (2.11) where From which it follows that there exists a constant , such that (2.12) where . Secondly, taking integration of the third equations in ( 2.7 ) over domain , we have (2.13) This implies that there exists a constant , such that (2.14) Let . Then (2.15) Moreover, there exists a positive constant which depends on and the -norm of , such that ( ) Step 2. estimate and . Multiplying the first three inequalities of Corollary 2.3 by , respectively, and integrating over , we have (2.16) Let . By the above three inequalities and Young inequality, we have (2.17) where (2.18) is quadratic form of . It is not hard to verify that is positive definite if ( 2.2 ) holds. Moreover, if ( 2.2 ) holds, then (2.19) Now we proceed in the following two cases. (i) It holds that . By ( 2.6 ) and ( 2.15 ), we have , and (2.20) By ( 2.19 ) and ( 2.20 ), we can see that (2.21) Thus, there exists positive constants and depending on , such that (2.22) Since the zero point of the right-hand side in ( 2.21 ) can be estimated by positive constants independent of , when . Thus do not depend on . (ii) . Repeating estimates in (i) by , we can obtain that there exists a positive constant depending on and the -norm of , such that ( ) when is independent of . Step 3. Estimate , . Introduce the scaling that (2.23) denote , and redenote by respectively. Then ( 2.7 ) reduces to (2.24) where . We still proceed in following two cases. (i) It holds that . From ( 2.15 ) and ( 2.22 ), we can easily obtain that (2.25) where . Multiply the first three equations in ( 2.24 ) by and integrate them over , respectively, then adding up the three new equations, we have (2.26) where . It is not hard to verify by ( 2.4 ) that there exists a positive constant depending only on , such that (2.27) Thus, (2.28) Using Young inequality, Hölder inequality and ( 2.24 ), we can obtain the following estimates: (2.29) Applying the above estimates and Gagliardo-Nirenberg-type inequalities to the terms on the right-hand side of ( 2.28 ), we have (2.30) Thus (2.31) For the other terms on the right-hand side of ( 2.28 ), we have (2.32) Thus (2.33) where is a positive constant. Note by ( 2.8 ) and ( 2.9 ) that , and (2.34) Choose a small enough number , such that According to ( 2.28 )–( 2.34 ), we have (2.35) where . However, ( 2.35 ) implies that there exist positive constants and depending on , such that (2.36) When , the coefficients of ( 2.35 ) can be estimated by constants depending on , but not on . Thus, when depends on , , and is irrelevant to . Since (2.37) similar to (2.26) in [ 24 ], we have (2.38) where is a positive constant only depending on . Scaling back with ( 2.22 ) to original variable and combining ( 2.36 ),( 2.38 ), there exist positive constants and depending on , such that (2.39) In addition, when is dependent of , but independent of . (ii)It holds that . Replacing with in ( 2.24 )–( 2.34 ), we can obtain that there exists a positive constant depending on and the -norm of such that ( ) When is dependent of , but independent of . Concluding from ( 2.15 ), ( 2.22 ), ( 2.39 ), and embedding theorem, there exists a positive constants depending on , such that ( 2.3 ) and ( 2.4 ) are satisfied. Furthermore, when and the time is large enough, are dependent of , but independent of . Similarly, according to ( ), ( ), ( ), we can see that there exists a positive constant depending on and the initial functions , such that (2.40) When is dependent of , but independent of . Thus . This completes proof of Theorem 2.1 . 3. Global Stability From [ 1 ], we know that if ( ) where , then ( 1.1 ) has the unique position equilibrium point . Theorem 3.1. Assume that all conditions in Theorem 2.1 and ( ) are satisfied. Assume further that (3.1) (3.2) hold, where is the positive constant in ( 2.4 ). Then the unique positive equilibrium point of ( 1.1 ) is globally asymptotically stable. Remark 3.2. Since is independent of in the case of , ( 3.2 ) is always satisfied if and are big enough. Proof. Define the Lyapunov function (3.3) Let be any solution of ( 1.1 ) with initial functions . From the strong maximum principle for parabolic equations, it is not hard to verify that for . Thus (3.4) The first integrand in the right hand of the above inequality is positive definite if (3.5) From the maximum-norm estimate in Theorem 2.1 , ( 3.2 ) is a sufficient condition of ( 3.5 ). Thus when ( 3.1 ) holds, there exists a positive constant such that (3.6) By integration by parts, Hölder inequality and the maximum-norm estimate in Theorem 2.1 , we can see that is bounded from above. According to Lemma in [ 1 ] and ( 3.6 ), we obtain (3.7) Using Gagliardo-Nirenberg inequalities, we have . Thus (3.8) That is, converges uniformly to . Since is decreasing for is globally asymptotically stable. <h4>Acknowledgments</h4> This work has been partially supported by the China National Natural Science Foundation (no. 10871160), the NSF of Gansu Province (no. 096RJZA118), the Scientific Research Fund of Gansu Provincial Education Department, and the NWNU-KJCXGC-03-47 Foundation. <h4>References</h4> R. Zhang, L. Guo, and S. Fu, “Global behavior for a diffusive predator-prey model with stage structure and nonlinear density restriction—I: the case in ℝ n ,” Boundary Value Problems , vol. 2009, Article ID 378763, p. 27, 2009. J. Blat and K. J. Brown, “Bifurcation of steady-state solutions in predator-prey and competition systems,” Proceedings of the Royal Society of Edinburgh , vol. 97, pp. 21–34, 1984. J. Blat and K. J. Brown, “ Global bifurcation of positive solutions in some systems of elliptic equations ,” SIAM Journal on Mathematical Analysis , vol. 17, no. 6, pp. 1339–1353, 1986. J. C. Eilbeck, J. E. Furter, and J. López-Gómez, “ Coexistence in the competition model with diffusion ,” Journal of Differential Equations , vol. 107, no. 1, pp. 96–139, 1994. E. N. Dancer, “ On positive solutions of some pairs of differential equations ,” Transactions of the American Mathematical Society , vol. 284, no. 2, pp. 729–743, 1984. E. N. Dancer, “ On positive solutions of some pairs of differential equations. II ,” Journal of Differential Equations , vol. 60, no. 2, pp. 236–258, 1985. E. N. Dancer, J. López-Gómez, and R. Ortega, “On the spectrum of some linear noncooperative elliptic systems with radial symmetry,” Differential and Integral Equations , vol. 8, no. 3, pp. 515–523, 1995. Y. Du and Y. Lou, “ Some uniqueness and exact multiplicity results for a predator-prey model ,” Transactions of the American Mathematical Society , vol. 349, no. 6, pp. 2443–2475, 1997. Y. Du and Y. Lou, “ S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model ,” Journal of Differential Equations , vol. 144, no. 2, pp. 390–440, 1998. Y. Du and Y. Lou, “ Qualitative behaviour of positive solutions of a predator-prey model: effects of saturation ,” Proceedings of the Royal Society of Edinburgh , vol. 131, no. 2, pp. 321–349, 2001. J. López-Gómez and R. M. Pardo, “Coexistence in a simple food chain with diffusion,” Journal of Mathematical Biology , vol. 30, no. 7, pp. 655–668, 1992. J. López-Gómez and R. M. Pardo, “Existence and uniqueness of coexistence states for the predator-prey model with diffusion: the scalar case,” Differential and Integral Equations , vol. 6, no. 5, pp. 1025–1031, 1993. J. López-Gómez and R. M. Pardo, “Invertibility of linear noncooperative elliptic systems,” Nonlinear Analysis: Theory, Methods & Applications , vol. 31, no. 5-6, pp. 687–699, 1998. K. Nakashima and Y. Yamada, “Positive steady states for prey-predator models with cross-diffusion,” Advances in Differential Equations , vol. 1, no. 6, pp. 1099–1122, 1996. H. Zhang, P. Georgescu, and L. Chen, “ An impulsive predator-prey system with Beddington-DeAngelis functional response and time delay ,” International Journal of Biomathematics , vol. 1, no. 1, pp. 1–17, 2008. Y. Fan, L. Wang, and M. Wang, “ Notes on multiple bifurcations in a delayed predator-prey model with nonmonotonic functional response ,” International Journal of Biomathematics , vol. 2, no. 2, pp. 129–138, 2009. F. Wang and Y. An, “ Existence of nontrivial solution for a nonlocal elliptic equation with nonlinear boundary condition ,” Boundary Value Problems , vol. 2009, Article ID 540360, 8 pages, 2009. S. Shim, “ Uniform boundedness and convergence of solutions to cross-diffusion systems ,” Journal of Differential Equations , vol. 185, no. 1, pp. 281–305, 2002. S. Shim, “ Uniform boundedness and convergence of solutions to the systems with cross-diffusions dominated by self-diffusions ,” Nonlinear Analysis: Real World Applications , vol. 4, no. 1, pp. 65–86, 2003. S. Shim, “ Uniform boundedness and convergence of solutions to the systems with a single nonzero cross-diffusion ,” Journal of Mathematical Analysis and Applications , vol. 279, no. 1, pp. 1–21, 2003. M. Wang, Nonliear Parabolic Equation of Parabolic Type , Science Press, Beijing, China, 1993. H. Amann, “Dynamic theory of quasilinear parabolic equations—II: reaction-diffusion systems,” Differential and Integral Equations , vol. 3, no. 1, pp. 13–75, 1990. H. Amann, “ Dynamic theory of quasilinear parabolic systems—III: global existence ,” Mathematische Zeitschrift , vol. 202, no. 2, pp. 219–250, 1989. S. Fu, Z. Wen, and S. Cui, “ Uniform boundedness and stability of global solutions in a strongly coupled three-species cooperating model ,” Nonlinear Analysis: Real World Applications , vol. 9, no. 2, pp. 272–289, 2008. // http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Boundary Value Problems Hindawi Publishing Corporation

Global Behavior for a Diffusive Predator-Prey Model with Stage Structure and Nonlinear Density Restriction-II: The Case in ℝ 1

Rui, Zhang; Ling, Guo; Shengmao, Fu
Boundary Value Problems , Volume 2009 (2009) – Dec 8, 2009
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Global Behavior for a Diffusive Predator-Prey Model with Stage Structure and Nonlinear Density Restriction-II: The Case in ℝ1 <meta name="citation_title" content="Global Behavior for a Diffusive Predator-Prey Model with Stage Structure and Nonlinear Density Restriction-II: The Case in ℝ 1 " /> //// Hindawi Publishing Corporation Home Journals About Us About this Journal Submit a Manuscript Table of Contents Journal Menu Abstracting and Indexing Aims and Scope Article Processing Charges Articles in Press Author Guidelines Bibliographic Information Contact Information Editorial Board Editorial Workflow Free eTOC Alerts Reviewers Acknowledgment Subscription Information Open Special Issues Published Special Issues Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Linked References How to Cite this Article Boundary Value Problems Volume 2009 (2009), Article ID 654539, 19 pages doi:10.1155/2009/654539 Research Article Global Behavior for a Diffusive Predator-Prey Model with Stage Structure and Nonlinear Density Restriction-II: The Case in ℝ 1 Rui Zhang , 1,2 Ling Guo , 1 and Shengmao Fu 1 1 Department of Mathematics, Northwest Normal University, Lanzhou 730070, China 2 Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China Received 2 April 2009; Accepted 31 August 2009 Academic Editor: Wenming Zou Copyright © 2009 Rui Zhang et al. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. <h4>Abstract</h4> A Holling type III predator-prey model with self- and cross-population pressure is considered. Using the energy estimate and Gagliardo-Nirenberg-type inequalities, the existence and uniform boundedness of global solutions to the model are dicussed. In addition, global asymptotic stability of the positive equilibrium point for the model is proved by Lyapunov function. 1. Introduction This paper is a continuation of Part I [ 1 ]. In Section of Part I, using the energy estimate and bootstrap arguments, the global existence of solutions for a Holling type III cross-diffusion predator-prey model with stage-structure has been discussed when the space dimension be less than 6. However, to obtain the estimate for the population density of predator species, there is not cross-diffusion for in Part I. All diffusive predator-prey systems behave, more or less, in the same way, for both semilinear and cross-diffusive models, at least for small values of the cross diffusivities. Consequently, all the available information for linear diffusive models is essential to realize the behavior of the most complicated cross-diffusive systems [ 2 – 17 ]. In this paper, we consider the following cross-diffusion system: (1.1) where and are positive constants. Also, are linear diffusion coefficients of , respectively, while are referred as self-diffusion pressures, and are cross-diffusion pressures. If , then ( 1.1 ) reduces to the system (1.4) of Part I. Recently, the work in [ 18 – 20 ] studied the existence, uniform boundedness, and uniform convergence of global solutions for the Lotka-Volterra cross-diffusion models without stage-structure in the case that the space dimension . In this paper, we consider mainly the existence and uniform boundedness of global solutions for the model ( 1.1 ) with nonlinear density restriction and stage-structure. Moreover, global asymptotic stability of the positive equilibrium point for ( 1.1 ) is proved by an important lemma of [ 21 ]. The proof is complete and complement the uniform convergence theorem in [ 18 – 20 ]. 2. Global Existence and Uniform Boundedness For simplicity, denote . The local existence result of solutions to ( 1.1 ) is an immediate consequence of a series of papers [ 22 , 23 ] by Amann. Roughly speaking, if , then ( 1.1 ) has a unique nonnegative solution , where is the maximal existence time for the solution. If satisfies (2.1) then . If, in addition, , then . The main result in this section is as follows. Theorem 2.1. Let is the unique nonnegative solution of ( 1.1 ) in its maximal existence interval . Assume that (2.2) Then there exists and positive constants which depend on , such that (2.3) (2.4) and . In particular, if , where and are positive constants, then depend on , but do not depend on . The following Gagliardo-Nirenberg-type inequalities and corresponding corollary play an importance role in the proof of Theorem 2.1 . Theorem 2.2 (see [ 18 ]). Let be a bounded domain with . For every function , the derivative satisfies the inequality (2.5) provided one of the following three conditions is satisfied: , or , and is not a nonnegative integer, where , for all , and the positive constant depends on . Corollary 2.3. There exists a positive constant such that (2.6) (2.7) (2.8) (2.9) For simplicity, denote that is Sobolev embedding constant or other kind of absolute constant. are some positive constants which depend on . Also, are positive constants which depend on . When do not depend on , but on . Proof of Theorem 2.1 . Step 1. Estimate , . Firstly, taking integration of the first and second equations in ( 2.7 ) over the domain , respectively, and combining the two integration equalities linearly, we have (2.10) From Young inequality and Hölder inequality, we can see (2.11) where From which it follows that there exists a constant , such that (2.12) where . Secondly, taking integration of the third equations in ( 2.7 ) over domain , we have (2.13) This implies that there exists a constant , such that (2.14) Let . Then (2.15) Moreover, there exists a positive constant which depends on and the -norm of , such that ( ) Step 2. estimate and . Multiplying the first three inequalities of Corollary 2.3 by , respectively, and integrating over , we have (2.16) Let . By the above three inequalities and Young inequality, we have (2.17) where (2.18) is quadratic form of . It is not hard to verify that is positive definite if ( 2.2 ) holds. Moreover, if ( 2.2 ) holds, then (2.19) Now we proceed in the following two cases. (i) It holds that . By ( 2.6 ) and ( 2.15 ), we have , and (2.20) By ( 2.19 ) and ( 2.20 ), we can see that (2.21) Thus, there exists positive constants and depending on , such that (2.22) Since the zero point of the right-hand side in ( 2.21 ) can be estimated by positive constants independent of , when . Thus do not depend on . (ii) . Repeating estimates in (i) by , we can obtain that there exists a positive constant depending on and the -norm of , such that ( ) when is independent of . Step 3. Estimate , . Introduce the scaling that (2.23) denote , and redenote by respectively. Then ( 2.7 ) reduces to (2.24) where . We still proceed in following two cases. (i) It holds that . From ( 2.15 ) and ( 2.22 ), we can easily obtain that (2.25) where . Multiply the first three equations in ( 2.24 ) by and integrate them over , respectively, then adding up the three new equations, we have (2.26) where . It is not hard to verify by ( 2.4 ) that there exists a positive constant depending only on , such that (2.27) Thus, (2.28) Using Young inequality, Hölder inequality and ( 2.24 ), we can obtain the following estimates: (2.29) Applying the above estimates and Gagliardo-Nirenberg-type inequalities to the terms on the right-hand side of ( 2.28 ), we have (2.30) Thus (2.31) For the other terms on the right-hand side of ( 2.28 ), we have (2.32) Thus (2.33) where is a positive constant. Note by ( 2.8 ) and ( 2.9 ) that , and (2.34) Choose a small enough number , such that According to ( 2.28 )–( 2.34 ), we have (2.35) where . However, ( 2.35 ) implies that there exist positive constants and depending on , such that (2.36) When , the coefficients of ( 2.35 ) can be estimated by constants depending on , but not on . Thus, when depends on , , and is irrelevant to . Since (2.37) similar to (2.26) in [ 24 ], we have (2.38) where is a positive constant only depending on . Scaling back with ( 2.22 ) to original variable and combining ( 2.36 ),( 2.38 ), there exist positive constants and depending on , such that (2.39) In addition, when is dependent of , but independent of . (ii)It holds that . Replacing with in ( 2.24 )–( 2.34 ), we can obtain that there exists a positive constant depending on and the -norm of such that ( ) When is dependent of , but independent of . Concluding from ( 2.15 ), ( 2.22 ), ( 2.39 ), and embedding theorem, there exists a positive constants depending on , such that ( 2.3 ) and ( 2.4 ) are satisfied. Furthermore, when and the time is large enough, are dependent of , but independent of . Similarly, according to ( ), ( ), ( ), we can see that there exists a positive constant depending on and the initial functions , such that (2.40) When is dependent of , but independent of . Thus . This completes proof of Theorem 2.1 . 3. Global Stability From [ 1 ], we know that if ( ) where , then ( 1.1 ) has the unique position equilibrium point . Theorem 3.1. Assume that all conditions in Theorem 2.1 and ( ) are satisfied. Assume further that (3.1) (3.2) hold, where is the positive constant in ( 2.4 ). Then the unique positive equilibrium point of ( 1.1 ) is globally asymptotically stable. Remark 3.2. Since is independent of in the case of , ( 3.2 ) is always satisfied if and are big enough. Proof. Define the Lyapunov function (3.3) Let be any solution of ( 1.1 ) with initial functions . From the strong maximum principle for parabolic equations, it is not hard to verify that for . Thus (3.4) The first integrand in the right hand of the above inequality is positive definite if (3.5) From the maximum-norm estimate in Theorem 2.1 , ( 3.2 ) is a sufficient condition of ( 3.5 ). Thus when ( 3.1 ) holds, there exists a positive constant such that (3.6) By integration by parts, Hölder inequality and the maximum-norm estimate in Theorem 2.1 , we can see that is bounded from above. According to Lemma in [ 1 ] and ( 3.6 ), we obtain (3.7) Using Gagliardo-Nirenberg inequalities, we have . Thus (3.8) That is, converges uniformly to . Since is decreasing for is globally asymptotically stable. <h4>Acknowledgments</h4> This work has been partially supported by the China National Natural Science Foundation (no. 10871160), the NSF of Gansu Province (no. 096RJZA118), the Scientific Research Fund of Gansu Provincial Education Department, and the NWNU-KJCXGC-03-47 Foundation. <h4>References</h4> R. Zhang, L. Guo, and S. Fu, “Global behavior for a diffusive predator-prey model with stage structure and nonlinear density restriction—I: the case in ℝ n ,” Boundary Value Problems , vol. 2009, Article ID 378763, p. 27, 2009. J. Blat and K. J. Brown, “Bifurcation of steady-state solutions in predator-prey and competition systems,” Proceedings of the Royal Society of Edinburgh , vol. 97, pp. 21–34, 1984. J. Blat and K. J. Brown, “ Global bifurcation of positive solutions in some systems of elliptic equations ,” SIAM Journal on Mathematical Analysis , vol. 17, no. 6, pp. 1339–1353, 1986. J. C. Eilbeck, J. E. Furter, and J. López-Gómez, “ Coexistence in the competition model with diffusion ,” Journal of Differential Equations , vol. 107, no. 1, pp. 96–139, 1994. E. N. Dancer, “ On positive solutions of some pairs of differential equations ,” Transactions of the American Mathematical Society , vol. 284, no. 2, pp. 729–743, 1984. E. N. Dancer, “ On positive solutions of some pairs of differential equations. II ,” Journal of Differential Equations , vol. 60, no. 2, pp. 236–258, 1985. E. N. Dancer, J. López-Gómez, and R. Ortega, “On the spectrum of some linear noncooperative elliptic systems with radial symmetry,” Differential and Integral Equations , vol. 8, no. 3, pp. 515–523, 1995. Y. Du and Y. Lou, “ Some uniqueness and exact multiplicity results for a predator-prey model ,” Transactions of the American Mathematical Society , vol. 349, no. 6, pp. 2443–2475, 1997. Y. Du and Y. Lou, “ S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model ,” Journal of Differential Equations , vol. 144, no. 2, pp. 390–440, 1998. Y. Du and Y. Lou, “ Qualitative behaviour of positive solutions of a predator-prey model: effects of saturation ,” Proceedings of the Royal Society of Edinburgh , vol. 131, no. 2, pp. 321–349, 2001. J. López-Gómez and R. M. Pardo, “Coexistence in a simple food chain with diffusion,” Journal of Mathematical Biology , vol. 30, no. 7, pp. 655–668, 1992. J. López-Gómez and R. M. Pardo, “Existence and uniqueness of coexistence states for the predator-prey model with diffusion: the scalar case,” Differential and Integral Equations , vol. 6, no. 5, pp. 1025–1031, 1993. J. López-Gómez and R. M. Pardo, “Invertibility of linear noncooperative elliptic systems,” Nonlinear Analysis: Theory, Methods & Applications , vol. 31, no. 5-6, pp. 687–699, 1998. K. Nakashima and Y. Yamada, “Positive steady states for prey-predator models with cross-diffusion,” Advances in Differential Equations , vol. 1, no. 6, pp. 1099–1122, 1996. H. Zhang, P. Georgescu, and L. Chen, “ An impulsive predator-prey system with Beddington-DeAngelis functional response and time delay ,” International Journal of Biomathematics , vol. 1, no. 1, pp. 1–17, 2008. Y. Fan, L. Wang, and M. Wang, “ Notes on multiple bifurcations in a delayed predator-prey model with nonmonotonic functional response ,” International Journal of Biomathematics , vol. 2, no. 2, pp. 129–138, 2009. F. Wang and Y. An, “ Existence of nontrivial solution for a nonlocal elliptic equation with nonlinear boundary condition ,” Boundary Value Problems , vol. 2009, Article ID 540360, 8 pages, 2009. S. Shim, “ Uniform boundedness and convergence of solutions to cross-diffusion systems ,” Journal of Differential Equations , vol. 185, no. 1, pp. 281–305, 2002. S. Shim, “ Uniform boundedness and convergence of solutions to the systems with cross-diffusions dominated by self-diffusions ,” Nonlinear Analysis: Real World Applications , vol. 4, no. 1, pp. 65–86, 2003. S. Shim, “ Uniform boundedness and convergence of solutions to the systems with a single nonzero cross-diffusion ,” Journal of Mathematical Analysis and Applications , vol. 279, no. 1, pp. 1–21, 2003. M. Wang, Nonliear Parabolic Equation of Parabolic Type , Science Press, Beijing, China, 1993. H. Amann, “Dynamic theory of quasilinear parabolic equations—II: reaction-diffusion systems,” Differential and Integral Equations , vol. 3, no. 1, pp. 13–75, 1990. H. Amann, “ Dynamic theory of quasilinear parabolic systems—III: global existence ,” Mathematische Zeitschrift , vol. 202, no. 2, pp. 219–250, 1989. S. Fu, Z. Wen, and S. Cui, “ Uniform boundedness and stability of global solutions in a strongly coupled three-species cooperating model ,” Nonlinear Analysis: Real World Applications , vol. 9, no. 2, pp. 272–289, 2008. //

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