A mathematical model for a spatial predator-prey interaction (original) (raw)
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Analysis of two generic spatially extended predator-prey models
2006
We present the analysis of two reaction-diusion systems modeling predator-prey interactions with the Holling Type II functional response and logistic growth of the prey. Initially we undertake the local analysis of the systems, deriving conditions on the parameters that guarantee a stable limit cycle in the reaction kinetics, and construct arbitrary large invariant regions in the equal diusion coecien t case. We then provide an a priori estimate that leads to the global well-posedness of the classical (nonnegative) solutions, given any nonnegative L 1 - initial data. In order to verify the biological wave phenomena of solutions and the theoretical results, numerical experiments are undertaken in two space dimensions using a Galerkin nite element method with piecewise linear continuous basis functions.
A diffusive predator–prey model in heterogeneous environment
Journal of Differential Equations, 2004
In this paper, we demonstrate some special behavior of steady-state solutions to a predatorprey model due to the introduction of spatial heterogeneity. We show that positive steadystate solutions with certain prescribed spatial patterns can be obtained when the spatial environment is designed suitably. Moreover, we observe some essential differences of the behavior of our model from that of the classical Lotka-Volterra model that seem to arise only in the heterogeneous case. Published by Elsevier Inc.
This paper deals with the study of a predator-prey model in a patchy environment. Prey individuals moves on two patches, one is a refuge and the second one contains predator individuals. The movements are assumed to be faster than growth and predator-prey interaction processes. Each patch is assumed to be homogeneous. The spatial heterogeneity is obtained by as- suming that the demographic parameters (growth rates, predation rates and mortality rates) depend on the patches. On the predation patch, we use a Lotka-Volterra model. Since the movements are faster that the other processes, we may assume that the frequency of prey and predators become constant and we would get a global predator-prey model, which is shown to be a Lotka-Volterra one. However, this simplified model at the population level does not match the dynamics obtained with the complete initial model. We explain this phenomenom and we continue the analysis in order to give a two-dimensional predator-prey model that give...
An Analysis of Some Models of Prey-predator Interaction
WSEAS transactions on biology and biomedicine, 2024
Biological models of basic prey-predator interaction have been studied. This consisted, at first, in analyzing the basic models of population dynamics such as the Malthus model, the Verhulst model, the Gompertz model and the model with Allee effect ; then, in a second step, to analyze the Lotka-Volterra model and its models improved by taking into account certain important hypotheses such as competition and/or cooperation between species, existence of refuge for prey and migration of species. For each population evolution model presented, a numerical illustration was made for its verification.
The modeling of predator-prey interactions
In this paper, we aim to study the interactions between the territorial animals like foxes and the rabbits. The territories for the foxes are considered to be the simple cells. The interactions between predator and its prey are represented by the chemical reactions which obey the mass action law. In this sense, we apply the mass action law for predator prey models and the quasi chemical approach is applied for the interactions between the predator and its prey to develop the modeled equations for different possible mechanisms of the predator prey interactions.
Mathematical study of multispecies dynamics modeling predator–prey spatial interactions
Journal of Numerical Mathematics, 2017
In this work, we present analysis of a scaled time-dependent reaction–diffusion system modeling three competitive species dynamics that is of Lotka–Volterra type for coexistence, permanence and stability. The linear analysis is based on the application of qualitative theory of ordinary differential equations and dynamical systems. We consider two notable spatial discretization methods in conjunction with an adaptive time stepping method to verify the biological wave phenomena of the solutions and present the numerical results in one dimensional space. Adequate numerical resulting are provided in one and two dimensions to justify theoretical investigations. In addition, efficiency of the proposed numerical schemes are justified.
Qualitative and numerical analysis of a class of prey-predator models
Acta Applicandae Mathematicae, 1985
We consider a problem of the dynamics of prey-predator populations suggested by the content of a letter of the biologist Umberto D'Ancona to Vito Volterra. The main feature of the problem is the special type of competition between predators of the same species as well as of different species. Two classes of cases are investigated: a first class in which the behaviour of the predator is ‘blind’ and the second one in which the behaviour is ‘intelligent’. A qualitative analysis of the dynamical systems under consideration is followed by a numerical analysis of the most significant cases.
Spatiotemporal dynamics of two generic predator–prey models
Journal of Biological Dynamics, 2010
We present the analysis of two reaction-diffusion systems modelling predator-prey interactions, where the predator displays the Holling type II functional response, and in the absence of predators, the prey growth is logistic. The local analysis is based on the application of qualitative theory for ordinary differential equations and dynamical systems, while the global well-posedness depends on invariant sets and differential inequalities. The key result is an L ∞-stability estimate, which depends on a polynomial growth condition for the kinetics. The existence of an a priori L p-estimate, uniform in time, for all p ≥ 1, implies L ∞uniform bounds, given any nonnegative L ∞-initial data. The applicability of the L ∞-estimate to general reaction-diffusion systems is discussed, and how the continuous results can be mimicked in the discrete case, leading to stability estimates for a Galerkin finite-element method with piecewise linear continuous basis functions. In order to verify the biological wave phenomena of solutions, numerical results are presented in two-space dimensions, which have interesting ecological implications as they demonstrate that solutions can be 'trapped' in an invariant region of phase space.
Impact of spatial heterogeneity on a predator–prey system dynamics
Comptes Rendus Biologies, 2004
This paper deals with the study of a predator-prey model in a patchy environment. Prey individuals moves on two patches, one is a refuge and the second one contains predator individuals. The movements are assumed to be faster than growth and predator-prey interaction processes. Each patch is assumed to be homogeneous. The spatial heterogeneity is obtained by assuming that the demographic parameters (growth rates, predation rates and mortality rates) depend on the patches. On the predation patch, we use a Lotka-Volterra model. Since the movements are faster that the other processes, we may assume that the frequency of prey and predators become constant and we would get a global predator-prey model, which is shown to be a Lotka-Volterra one. However, this simplified model at the population level does not match the dynamics obtained with the complete initial model. We explain this phenomenom and we continue the analysis in order to give a two-dimensional predator-prey model that gives the same dynamics as that provided by the complete initial one. We use this simplified model to study the impact of spatial heterogeneity and movements on the system stability. This analysis shows that there is a globally asymptotically stable equilibrium in the positive quadrant, i.e. the spatial heterogeneity stabilizes the equilibrium. To cite this article:
Global dynamics of a predator–prey model
Journal of Mathematical Analysis and Applications, 2010
This paper deals with the dynamics of a predator-prey model with Hassell-Varley-Holling functional response. First, we show that the predator coexists with prey if and only if predator's growth ability is greater than its death rate. Second, using a blow-up technique, we prove that the origin equilibrium point is repelling and extinction of both predator and prey populations is impossible. Third, the local and global stability of the positive steady state coincide when the predator interference is large. Finally, for a typical biological case, we show instability of the positive equilibrium implies global stability of the limit cycle. Numerical simulations are carried out for a hypothetical set of parameter values to substantiate our analytical findings.