Predators–prey models with competition, Part I: Existence, bifurcation and qualitative properties

Abstract

We study a mathematical model of environments populated by both preys and predators, with the possibility for predators to actively compete for the territory. For this model we study existence and uniqueness of solutions, and their asymptotic properties in time, showing that the solutions have different behavior depending on the choice of the parameters. We also construct heterogeneous stationary solutions and study the limits of strong competition and abundant resources. We then use these information to study some properties such as the existence of solutions that maximize the total population of predators. We prove that in some regimes the optimal solution for the size of the total population contains two or more groups of competing predators.the dimension is purely for modeling reasons) occupied by N + 1 groups of animals. The density of the first one, which we denote as u, of preys while the remaining N densities, denoted 2010 Mathematics Subject Classification. Primary: 35Q92; secondary: 35A01, 35B25, 35B32, 92D50.

Some remarks on segregation of $k$ species in strongly competing systems

“…From a mathematical viewpoint the determination of the configuration of the habitat segregation for some populations is an interesting problem which can be modelled by an optimal (in a suitable sense) partition of a domain; for example in the papers [8][9][10][11]22] the problem is studied modelling the interspecies competition with a large interaction term in an elliptic system of partial differential equations inspired by classical models in populations dynamics. In [3,4,12] the problem is modelled as a Cauchy problem for a parabolic system of semilinear partial differential equations describing the dynamics of the densities of different species. In the evolutive case, in particular see [12], it is proved that some populations can vanish under the competition of other species; moreover, in [3,4] the authors are able to estimate the number of the long-term surviving populations and other interesting qualitative properties of the spatial distributions of interacting populations.…”

“…In [3,4,12] the problem is modelled as a Cauchy problem for a parabolic system of semilinear partial differential equations describing the dynamics of the densities of different species. In the evolutive case, in particular see [12], it is proved that some populations can vanish under the competition of other species; moreover, in [3,4] the authors are able to estimate the number of the long-term surviving populations and other interesting qualitative properties of the spatial distributions of interacting populations. Note that also the study of the territoriality, that is how different groups of the same species divide an area, avoiding to effectively fight for resources, can be viewed as a habitat segregation produced by competition (see, for example, [5,16,20]); moreover, this kind of competition is a struggle between competitors having the same features, that is between perfect competitors.…”

Qualitative features of a nonlinear, nonlocal, agent-based PDE model with applications to homelessness

“…4,14,26 Species aggregation has also been thoroughly studied, 7 with many authors modeling such phenomena by nonlocal differential equations. 2,10,46 On even larger scales, we can see the natural emergence of pack formation in predator-prey systems, for sufficiently strong intraspecies competition; 6 descriptions of human territorial conquest; 17 and ecology models for a species in an environment with patchy resources. 34 But to our knowledge, little has been done directly to model homelessness, which is a growing societal concern.…”

On the limit configuration of four species strongly competing systems

“…From a mathematical viewpoint the determination of the configuration of the habitat segregation for some populations is an interesting problem which can be modelled by an optimal (in a suitable sense) partition of a domain, for example in the papers [4,5,6,7,14] the problem is studied modelling the interspecies arXiv:1809.10159v2 [math.AP] 14 May 2019 competition with a large interaction term in an elliptic system of partial differential equations inspired by classical models in populations dynamics. In [3,8] the problem is modelled as a Cauchy problem for a parabolic system of semilinear partial differential equations describing the dynamics of the densities of different species. Starting from the quoted papers we want to tackle the segregation problem of 4 species in a planar region.…”

“…Starting from the quoted papers we want to tackle the segregation problem of 4 species in a planar region. Note that in the evolution works, in particular [8], it is proved that some populations can vanish under the competition of other species, moreover in [3] the authors are able to estimate the number of the long-term surviving populations.…”