Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey

Banerjee, Malay; Mukherjee, Nayana; Volpert, Vitaly

doi:10.3390/math6030041

Cited by 17 publications

(16 citation statements)

References 52 publications

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“…They further showed that the incorporation of nonlocal interactions in the classical Rosenzweig-MacArthur model can produce Turing patterns, while the local counterpart of it is unable to do so [47]. Some other interesting works on the impacts of nonlocal interactions on spatiotemporal dynamics for two-species models can be found in [40,[48][49][50][51][52][53][54].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species

2020

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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 prey species. Nonlocalities are introduced in the intra-specific competition terms for the two prey species in order to model the accessibility of nearby resources. Using linear analysis, we have derived the Turing instability conditions for both the spatiotemporal models with and without nonlocal interactions. Validation of such conditions indicates the possibility of existence of stationary spatially heterogeneous distributions for all the three species. Existence of long transient dynamics has been presented under certain parametric domain. Exhaustive numerical simulations reveal various scenarios of stabilization of population distribution due to the presence of nonlocal intra-specific competition for the two prey species. Chaotic oscillation exhibited by the temporal model is significantly suppressed when the populations are allowed to move over their habitat and prey species can access the nearby resources.

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Section: Introductionmentioning

confidence: 99%

Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species

2020

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“…Indeed, this area of research has proven to be extremely fruitful in view of the wide range of possible scenarios that merit investigation. As examples, we can mention studies that report on the modeling and analysis of predator-prey models with disease in the prey [1], the analysis of stochastic systems with modified Leslie-Gower and Holling-type schemes [2], the dynamic behaviors of Lotka-Volterra predator-prey models that incorporate predator cannibalism [3], the analysis of diffusive predator-prey systems with Michaelis-Menten-type predator harvesting [4], synthetic Escherichia coli predator-prey ecosystems [5], the analytical investigation of stage-structured predator-prey models depending on maturation delay and death rate [6], and non-autonomous ratio dependent models with Holling-type functional response with temporal delay [7], among other interesting topics [8].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion

Alba-Pérez

Macías‐Díaz

2019

Mathematics

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In this work, we investigate numerically a system of partial differential equations that describes the interactions between populations of predators and preys. The system considers the effects of anomalous diffusion and generalized Michaelis–Menten-type reactions. For the sake of generality, we consider an extended form of that system in various spatial dimensions and propose two finite-difference methods to approximate its solutions. Both methodologies are presented in alternative forms to facilitate their analyses and computer implementations. We show that both schemes are structure-preserving techniques, in the sense that they can keep the positive and bounded character of the computational approximations. This is in agreement with the relevant solutions of the original population model. Moreover, we prove rigorously that the schemes are consistent discretizations of the generalized continuous model and that they are stable and convergent. The methodologies were implemented efficiently using MATLAB. Some computer simulations are provided for illustration purposes. In particular, we use our schemes in the investigation of complex patterns in some two- and three-dimensional predator–prey systems with anomalous diffusion.

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“…In the absence of predator (α = 0) dynamics of prey is described by a reaction-diffusion equation with bistable reaction kinetics. The conditions for Turing instability and spatial Hopf bifurcation are explicitly derived and validated with numerical examples in [6]. The nonlocal model is capable of producing stationary Turing pattern, periodic traveling wave, modulated traveling wave in addition to oscillatory solutions and spatio-temporal chaos produced by the model without the nonlocal term.…”

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confidence: 97%

“…In a similar manner, when a prey-predator model with bistable reaction kinetics for prey (in the absence of predators) is considered, and nonlocal consumption of resources by prey is included into the system, various spatio-temporal patterns are reported in [6]. The model has the form:…”

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confidence: 99%

Prey-predator model with nonlocal and global consumption in the prey dynamics

Banerjee¹,

Mukherjee²,

Volpert³

2020

Discrete &Amp; Continuous Dynamical Systems - S

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A prey-predator model with a nonlocal or global consumption of resources by prey is studied. Linear stability analysis about the homogeneous in space stationary solution is carried out to determine the conditions of the bifurcation of stationary and moving pulses in the case of global consumption. Their existence is confirmed in numerical simulations. Periodic travelling waves and multiple pulses are observed for the nonlocal consumption.2010 Mathematics Subject Classification. Primary: 35K57.

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Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey

Cited by 17 publications

References 52 publications

Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species

Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species

Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion

Prey-predator model with nonlocal and global consumption in the prey dynamics

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