A nonlocal swarm model for predators–prey interactions (original) (raw)
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Journal of Mathematical Biology, 1999
This paper describes continuum models for swarming behavior based on non-local interactions. The interactions are assumed to in#uence the velocity of the organisms. The model consists of integro-di!erential advection-di!usion equations, with convolution terms that describe long range attraction and repulsion. We "nd that if density dependence in the repulsion term is of a higher order than in the attraction term, then the swarm pro"le is realistic: i.e. the swarm has a constant interior density, with sharp edges, as observed in biological examples. This is our main result. Linear stability analysis, singular perturbation theory, and numerical experiments reveal that weak, density-independent di!usion leads to disintegration of the swarm, but only on an exponentially large time scale. When density dependence is put into the di!usion term, we "nd that true, locally stable traveling band solutions occur. We further explore the e!ects of local and non-local density dependent drift and unequal ranges of attraction and repulsion. We compare our results with results of some local models, and "nd that such models cannot account for cohesive, "nite swarms with realistic density pro"les.
A nonlocal kinetic model for predator-prey interactions in two dimensions
2012
Eulerian models based on integro-differential equations may be used to model collective behaviour, by treating the group of individuals as a population density. In comparison with Lagrangian models, where one tracks distinct individuals, Eulerian models are formulated as evolution equations for the density field, and hence permit rigorous analysis to be performed. The population densities are influenced by the social interactions of attraction, repulsion and alignment. We introduce a new model for predator-prey dynamics that generalizes a previous integro-differential equation model by introducing the predator dynamics and a blind zone for the prey. Extensive simulations were performed to showcase the realism of the model, and these simulations are presented in four stages.
A predator-prey reaction-diffusion system with nonlocal effects
Journal of Mathematical Biology, 1996
Weconsider apredator-prey systemin theform of a coupled system of reaction-diffusion equations with an integral term representing a weighted average of the values of the prey density function, both in past time and space. In a limiting case the system reduces to the Lotka Volterra diffusion system with logistic growth of the prey. We investigate the linear stability of the coexistence steady state and bifurcations occurring from it, and expressions for some of the bifurcating solutions are constructed. None of these bifurcations can occur in the degenerate case when the nonlocal term is in fact local.
Non-local reaction-diffusion equations modelling predator-prey coevolution
Publicacions Matemàtiques, 1994
In this paper we examine a prey-predator system with a characteristic of the predator subject to mutation. The ultimate equilibrium of the system is found by Maynard-Smith et al. by th e so called ESS (Evolutionary Stable Strategy). Using a system of reaction-diffusion equations with non local terms, we conclud e the ESS result for the diffusion coefficient tending to zero, without resorting to any optimization criterion .
Stabilization of a Predator-Prey System with Nonlocal Terms
Mathematical Modelling of Natural Phenomena, 2015
We investigate the zero-stabilizability for the prey population in a predator-prey system via a control which acts in a subregion ω of the habitat Ω, and on the predators only. The dynamics of both interacting populations is described by a reaction-diffusion system with nonlocal terms describing migrations. A necessary condition and a sufficient condition for the zero-stabilizability of the prey population are derived in terms of the sign of the principal eigenvalues to certain non-selfadjoint operators. In case of stabilizability, a constant stabilizing control is indicated. The rate of stabilization corresponding to such a stabilizing control is dictated by the principal eigenvalue of a certain operator. A large principal eigenvalue leads to a fast stabilization to zero of the prey population. A method to approximate all these principal eigenvalues is presented. Some final comments concerning the relationship between the stabilization rate and the properties of ω and Ω are given as well.
Nonlocal generalized models of predator-prey systems
Discrete and Continuous Dynamical Systems - Series B, 2012
The method of generalized modeling has been applied successfully in many different contexts, particularly in ecology and systems biology. It can be used to analyze the stability and bifurcations of steady-state solutions. Although many dynamical systems in mathematical biology exhibit steady-state behaviour one also wants to understand nonlocal dynamics beyond equilibrium points. In this paper we analyze predatorprey dynamical systems and extend the method of generalized models to periodic solutions. First, we adapt the equilibrium generalized modeling approach and compute the unique Floquet multiplier of the periodic solution which depends upon so-called generalized elasticity and scale functions. We prove that these functions also have to satisfy a flow on parameter (or moduli) space. Then we use Fourier analysis to provide computable conditions for stability and the moduli space flow. The final stability analysis reduces to two discrete convolutions which can be interpreted to understand when the predator-prey system is stable and what factors enhance or prohibit stable oscillatory behaviour. Finally, we provide a sampling algorithm for parameter space based on nonlinear optimization and the Fast Fourier Transform which enables us to gain a statistical understanding of the stability properties of periodic predator-prey dynamics.
Mathematics, 2020
Investigation of interacting populations is an active area of research, and various modeling approaches have been adopted to describe their dynamics. Mathematical models of such interactions using differential equations are capable to mimic the stationary and oscillating (regular or irregular) population distributions. Recently, some researchers have paid their attention to explain the consequences of transient dynamics of population density (especially the long transients) and able to capture such behaviors with simple models. Existence of multiple stationary patches and settlement to a stable distribution after a long quasi-stable transient dynamics can be explained by spatiotemporal models with nonlocal interaction terms. However, the studies of such interesting phenomena for three interacting species are not abundant in literature. Motivated by these facts here we have considered a three species prey–predator model where the predator is generalist in nature as it survives on two...
A Nonlocal Continuum Model for Biological Aggregation
Bulletin of Mathematical Biology, 2006
We construct a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. For the case of one spatial dimension, we study the steady states analytically and numerically. There exist strongly nonlinear states with compact support and steep edges that correspond to localized biological aggregations, or clumps. These steady state clumps are approached through a dynamic coarsening process. In the limit of large population size, the clumps approach a constant density swarm with abrupt edges. We use energy arguments to understand the nonlinear selection of clump solutions, and to predict the internal density in the large population limit. The energy result holds in higher dimensions, as well, and is demonstrated via numerical simulations in two dimensions.
Mathematical Models and Methods in Applied Sciences, 2018
Hydrodynamic systems arising in swarming modeling include nonlocal forces in the form of attractive–repulsive potentials as well as pressure terms modeling strong local repulsion. We focus on the case where there is a balance between nonlocal attraction and local pressure in presence of confinement in the whole space. Under suitable assumptions on the potentials and the pressure functions, we show the global existence of weak solutions for the hydrodynamic model with viscosity and linear damping. By introducing linear damping in the system, we ensure the existence and uniqueness of stationary solutions with compactly supported density, fixed mass and center of mass. The associated velocity field is zero in the support of the density. Moreover, we show that global weak solutions converge for large times to the set of these stationary solutions in a suitable sense. In particular cases, we can identify the limiting density uniquely as the global minimizer of the free energy with the ri...
Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator
Discrete & Continuous Dynamical Systems - S, 2018
In this paper, we study the spatiotemporal dynamics of a diffusive predator-prey model with generalist predator subject to homogeneous Neumann boundary condition. Some basic dynamics including the dissipation, persistence and non-persistence(i.e., one species goes extinct), the local and global stability of non-negative constant steady states of the model are investigated. The conditions of Turing instability due to diffusion at positive constant steady states are presented. A critical value ρ of the ratio d 2 d 1 of diffusions of predator to prey is obtained, such that if d 2 d 1 > ρ, then along with other suitable conditions Turing bifurcation will emerge at a positive steady state, in particular so it is with the large diffusion rate of predator or the small diffusion rate of prey; while if d 2 d 1 < ρ, both the reaction-diffusion system and its corresponding ODE system are stable at the positive steady state. In addition, we provide some results on the existence and non-existence of positive non-constant steady states. These existence results indicate that the occurrence of Turing bifurcation, along with other suitable conditions, implies the existence of non-constant positive steady states bifurcating from the constant solution. At last, by numerical simulations, we demonstrate Turing pattern formation on the effect of the varied diffusive ratio d 2 d 1. As d 2 d 1 increases, Turing patterns change from spots pattern, stripes pattern into spots-stripes pattern. It indicates that the pattern formation of the model is rich and complex.