Existence and non-existence of spatial patterns in a ratio-dependent predator–prey model

Banerjee, Malay; Abbas, Syed

doi:10.1016/j.ecocom.2014.05.005

Cited by 48 publications

(23 citation statements)

References 65 publications

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“…Chaotic nature of the patterns presented in Figure 18a,b can be easily observed from the phase portraits illustrated in Figure 19. The chaotic nature of these patterns have also been cross verified by checking the sensitivity to initial conditions as described in [59], but we do not present the results here for the sake of brevity. For the intermediate periodic in both the space and time population distributions presented in Figures 5b and 8b, we found that the small extents of nonlocality for both the prey species could potentially stabilize the population distributions by giving rise to the stationary periodic in space distributions and we chose not to display these results here for the sake of brevity.…”

Section: Numerical Resultsmentioning

confidence: 89%

“…A close look at the chaotic pattern reveals that the distribution of the species is symmetric about the midpoint in space and this is due to our consideration of the symmetric pulse type initial conditions. The chaotic nature has been cross verified by checking the sensitivity to initial conditions as described in [59], but we do not present the results here for the sake of brevity. Interestingly, spatiotemporal chaotic pattern disappears and eventually settles to a stationary pattern for a higher value of d 3 (for example, d 3 = 10.0).…”

Section: Numerical Resultsmentioning

confidence: 99%

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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: Numerical Resultsmentioning

confidence: 89%

Section: Numerical Resultsmentioning

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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“…The present work is based upon a Michaelis-Menten type prey-predator model with ratio-dependent functional response and density dependent death rate of predator [2]. In terms of the dimensionless variables and parameters, the basic temporal model is governed by the following system of nonlinear coupled ordinary differential equations…”

Section: Basic Modelmentioning

confidence: 99%

“…All of them are re-scaled parameters. The model (2.1) exhibits a wide variety of local and global bifurcations depending upon the parametric restrictions [2]. Here we just mention the parametric conditions required for the feasible existence of interior equilibrium point and the conditions for Hopf-bifurcation without going into the details.…”

Section: Basic Modelmentioning

confidence: 99%

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Stabilizing Role of Nonlocal Interaction on Spatio-temporal Pattern Formation

Banerjee

Zhang

2016

Math. Model. Nat. Phenom.

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Here we study a spatio-temporal prey-predator model with ratio-dependent functional response and nonlocal interaction term in the prey growth. For a clear understanding of the effect of nonlocal interaction on the resulting stationary and non-stationary patterns, we consider the nonlocal interaction term in prey growth only to describe the nonlocal intra-specific competition due to limited resources for the prey. First we obtain the patterns exhibited by the basic model in the absence of nonlocal interaction and then explore the effect of nonlocal interaction on the resulting patterns. We demonstrate the stabilizing role of nonlocal interaction as it induces stationary pattern from periodic and chaotic regimes with an increase in the range of nonlocal interaction. The existence of multiple branches of stationary solutions, bifurcating from homogeneous steady-state as well as non-stationary patterns, is illustrated with the help of numerical continuation technique.

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