ISSN 0005-1179, Automation and Remote Control, 2017, Vol. 78, No. 8, pp. 1500–1511.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
A.S. Ivanova, A.N. Kirillov, 2015, published in Upravlenie Bol’shimi Sistemami, 2015, No. 55, pp. 239–258.
LARGE SCALE SYSTEMS CONTROL
Equilibrium and Control
in the Biocommunity Species
Composition Preservation Problem
A. S. Ivanova
and A. N. Kirillov
Institute of Applied Mathematical Research,
Karelian Research Center of the Russian Academy of Sciences, Petrozavodsk, Russia
a s firstname.lastname@example.org,
Received February 6, 2015
Abstract—This paper proposes mathematical models for the biocommunity species composition
preservation problem. We construct equilibrium for the dynamical model describing the self-
regulation of populations in a patch. For the model with the varying food attractivity of the
patch, we design specimen removal control that allows to preserve the species composition.
Foraging theory proceeds from the assumption that food resources consumed by a population are
distributed over patches. To put it tentatively, a population solves two problems, namely, chooses
an appropriate patch and the right time for leaving it (in case of food deﬁciency). Numerous
publications were dedicated to this branch of mathematical ecology [3, 5, 6, 9, 10]. In his classical
work , E. Charnov introduced a static model yielding the conditions under which a population
leaves a patch, known as the marginal value theorem. The present paper proposes and analyzes
the dynamic models of leaving a patch by a population in the case of insuﬃcient food attractivity.
In what follows, we suggest two dynamical models of leaving a patch by a population of some
species. The possibility to preserve the species composition of this patch is studied. Interaction
within the patch is described by the Lotka–Volterra system. Migration in the ﬁrst model is incorpo-
rated into the predator-prey dynamic equations. The second model represents a variable structure
system and migration there is deﬁned by a separate system. For the ﬁrst model, we calculate the
Nash equilibrium reﬂecting the self-regulation of populations in the patch. For the second model,
we design specimen removal control to preserve the species composition of the patch under the
assumption that only the predator population can migrate. The food attractivity of the patch for
the predator population depends on the number of prey specimens per one predator specimen and
varies in time. Moreover, the model describes predator’s return to the patch, i.e., can be used to
solve the problem of patch choice by it.
2. EQUILIBRIUM IN TWO-SPECIES MODEL WITH MIGRATION
Consider the Lotka–Volterra system with migration
(a − bq
) − μ
− m)− μ
are the quantitative characteristics (sizes) of the prey and predator populations,
respectively; a denotes the growth rate of the prey population; b gives the prey consumption rate