A nonlocal kinetic model for predator-prey interactions in two dimensions (original) (raw)
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A nonlocal swarm model for predators–prey interactions
Mathematical Models and Methods in Applied Sciences, 2016
We consider a two-species system of nonlocal interaction PDEs modeling the swarming dynamics of predators and prey, in which all agents interact through attractive/repulsive forces of gradient type. In order to model the predator–prey interaction, we prescribed proportional potentials (with opposite signs) for the cross-interaction part. The model has a particle-based discrete (ODE) version and a continuum PDE version. We investigate the structure of particle stationary solution and their stability in the ODE system in a systematic form, and then consider simple examples. We then prove that the stable particle steady states are locally stable for the fully nonlinear continuum model, provided a slight reinforcement of the particle condition is required. The latter result holds in one space dimension. We complement all the particle examples with simple numerical simulations, and we provide some two-dimensional examples to highlight the complexity in the large time behavior of the system.
Directed Movement of Predators and the Emergence of Density-Dependence in Predator–Prey Models
Theoretical Population Biology, 2001
We consider a bitrophic spatially distributed community consisting of prey and actively moving predators. The model is based on the assumption that the spatial and temporal variations of the predators' velocity are determined by the prey gradient. Locally, the populations follow the simple Lotka Volterra interaction. We also assume predator reproduction and mortality to be negligible in comparison with the time scale of migration. The model demonstrates heterogeneous oscillating distributions of both species, which occur because of the active movements of predators. One consequence of this heterogeneity is increased viability of the prey population, compared to the equivalent homogeneous model, and increased consumption. Further numerical analysis shows that, on the spatially aggregated scale, the average predator density adversely affects the individual consumption, leading to a nonlinear predatordependent trophic function, completely different from the Lotka Volterra rule assumed at the local scale. ]
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 mathematical model for a spatial predator-prey interaction
Mathematical Methods in the Applied Sciences, 2002
A predator-prey model is proposed within the general scheme of extended thermodynamics. An additional equation of balance is needed to account for the ux of the number density of the predator as an independent ÿeld variable.
Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey
Mathematics, 2018
Spatiotemporal pattern formation in integro-differential equation models of interacting populations is an active area of research, which has emerged through the introduction of nonlocal intra-and inter-specific interactions. Stationary patterns are reported for nonlocal interactions in prey and predator populations for models with prey-dependent functional response, specialist predator and linear intrinsic death rate for predator species. The primary goal of our present work is to consider nonlocal consumption of resources in a spatiotemporal prey-predator model with bistable reaction kinetics for prey growth in the absence of predators. We derive the conditions of the Turing and of the spatial Hopf bifurcation around the coexisting homogeneous steady-state and verify the analytical results through extensive numerical simulations. Bifurcations of spatial patterns are also explored numerically.
Dynamics of prey-flock escaping behavior in response to predator's attack
Journal of Theoretical Biology, 2006
The dynamic behavior of prey-flock in response to predator's attack was investigated by using molecular dynamics (MD) simulations in a two-dimensional (2D) continuum model. By locally applying interactive forces between prey individuals (e.g. attraction, repulsion, and alignment), a coherently moving state in the same direction was obtained among individuals in prey-flock. When a single predator was introduced to the prey population, the prey-flock was correspondingly deformed by the predator's continuous attacks towards the prey-flock's center. In response to the predator's attack, three regimes in the flock size (compression (Regime I), expansion (Regime II), compression (Regime III)) were revealed if the predator's attack speed ðkÞ was comparatively low to the escape speed of prey-flock. If noise was added to the predator's attacking course, a higher degree of variation was observed in the patterns of compression and expansion in the prey-flock size. However, the scaling behavior in the changes in prey-flock size was present in different levels of noise with the increase in predation risk (R) when k takes an appropriately low value. During the procedure of escaping, order breaking in alignment (f) of prey population was observed, while the degree of alignment was dependent upon the changes in parameters of k and R.
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.
Journal of Theoretical Biology, 1995
The Lotka-Volterra model of predator-prey interaction is based on the assumption of mass action, a concept borrowed from the traditional theory of chemical kinetics in which reactants are assumed to be homogeneously mixed. In order to explore the effect of spatial heterogeneity on predator-prey dynamics, we constructed a lattice-based reaction-diffusion model corresponding to the Lotka-Volterra equations. Spatial heterogeneity was imposed on the system using percolation maps, gradient percolation maps, and fractional Brownian surfaces. In all simulations where diffusion distances were short, anomalously low reaction orders and aggregated spatial patterns were observed, including traveling wave patterns. In general, the estimated reaction order decreased with increasing degrees of spatial heterogeneity. For simulations using percolation maps with p-values varying between 1.0 (all cells available) to 0.5 (50% available), order estimates varied from 1.27 to 0.47. Gradient percolation maps and fractional Brownian surfaces also resulted in anomalously low reaction orders. Increasing diffusion distances resulted in reaction order estimates approaching the expected value of 2. Analysis of the qualitative dynamics of the model showed little difference between simulations where individuals diffused locally and those where individuals moved to random locations, suggesting that global density dependence is an important determinant of the overall model dynamics. However, localized interactions did introduce time dependence in the system attractor owing to emergent spatial patterns. We conclude that individual-based spatially explicit models are important tools for modeling population dynamics as they allow one to incorporate fine-scale ecological data about localized interactions and then to observe emergent patterns through simulation. When heterogeneous patterns arise, it can lead to anomalies with respect to the predictions of traditional mathematical approaches using global state variables.
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...
Predator-prey models in heterogeneous environment: Emergence of functional response
Mathematical and Computer Modelling, 1998
In this work, we are interested in prey-predator models. More precisely, we study the spatial heterogeneity effects on the amount of prey eaten per predator per unit time, when different time scales occur. This amount and its relation with the amount of predators produced via the predation are interesting from an ecological point of view. Indeed, the knowledge of these quantities permits us to quantify the transfer of the biomass in the food chain. Our aim is to show how the spatial heterogeneity acts on these amounts. We consider prey-predator systems in a multi-patch environment. We show that density dependent migrations make emerge new models on the total population level and we exhibit some examples. Furthermore, we show that the aggregation method is a good tool for describing the mechanisms hidden behind complex models.