# Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey

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## Abstract

**:**

## 1. Introduction

## 2. Stability Analysis

#### 2.1. Local Model

#### 2.2. Nonlocal Model

#### 2.3. Spatial Hopf Bifurcation

#### 2.4. Turing Pattern for Nonlocal Prey-Predator Model

## 3. Spatiotemporal Patterns

#### 3.1. Patterns Produced by the Model (8)–(9)

#### 3.2. Effect of Nonlocal Consumption

#### Multiplicity of Stationary Solutions

## 4. Discussion

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Turing bifurcation curve on the (M, ${d}_{2}$)-parameter plane (

**a**). Stationary Turing patterns exist above the bifurcation curve. The functions $\Delta $(k, 12.5) (

**b**) and $\Delta $(k, 12.7) (

**c**).The root corresponding to the Turing instability is shown in green.

**Figure 3.**(

**a**) Resulting spatio-temporal patterns produced by the nonlocal model (13)–(14) for ${d}_{2}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.3$, $M\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}6.5$ and other parameter values as mentioned in the text. (

**b**) Stationary pattern produced by the prey population for ${d}_{1}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1$, ${d}_{2}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1$ and $M\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}6$, other parameters are mentioned at text.

**Figure 4.**Spatiotemporal patterns (prey density) produced by the model (8)–(9) are presented in the left column for the parameter values as mentioned at the text and (

**a**) $\beta \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.342$; (

**c**) $\beta \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.3445$; (

**e**) $\beta \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.36$. Corresponding distribution of prey and predator population over space at $t\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1000$ are presented in the right column, see (

**b**), (

**d**), (

**f**).

**Figure 5.**Bifurcation diagram in $(M,\phantom{\rule{0.166667em}{0ex}}\beta )$-parameter space for fixed parameters $a\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1$, $b\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1$, ${\sigma}_{1}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.1$, $\alpha \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.4$, $\kappa \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.4$ and ${\sigma}_{2}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.2$. (

**a**) ${d}_{1}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1$, ${d}_{2}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1$; (

**b**) ${d}_{1}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.7$, ${d}_{2}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.5$.

**Figure 6.**Spatio-temporal patterns produced by the nonlocal model with parameter values as mentioned at (1) with $\beta \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.342$ and (

**a**) $M\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0$; (

**b**) $M\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}4$; (

**c**) $M\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}6$.

**Figure 7.**Stationary patterns with various number of patches exist for a range of nonlocal consumption (M) is plotted. Parameter values are same as in (24) except ${d}_{1}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.4$ and ${d}_{2}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.2$.

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Banerjee, M.; Mukherjee, N.; Volpert, V. Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey. *Mathematics* **2018**, *6*, 41.
https://doi.org/10.3390/math6030041

**AMA Style**

Banerjee M, Mukherjee N, Volpert V. Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey. *Mathematics*. 2018; 6(3):41.
https://doi.org/10.3390/math6030041

**Chicago/Turabian Style**

Banerjee, Malay, Nayana Mukherjee, and Vitaly Volpert. 2018. "Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey" *Mathematics* 6, no. 3: 41.
https://doi.org/10.3390/math6030041