# Predator–Prey Dynamics and Ideal Free Distribution in a Heterogeneous Environment

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations for Predator–Prey Dynamics in a Heterogeneous Habitat

## 3. Local Interaction (Main System without Fluxes)

**Proposition**

**1.**

**Proof.**

## 4. Ideal Free Distribution and Its Advancement

**Proposition**

**2.**

**Proof.**

## 5. Computer Experiment with IFD and Its Extension

## 6. Violation of Ideal Free Distribution

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IFD | Ideal Free Distribution |

ODE | Ordinary Differential Equation |

## Appendix A. Numerical Scheme

## References

- Murray, J.D. Mathematical Biology; Springer: New York, NY, USA, 2003. [Google Scholar]
- Cosner, C.; Cantrell, R. Spatial Ecology via Reaction—Diffusion Equations; John Wiley and Sons Ltd.: Chichester, UK, 2003. [Google Scholar]
- Fretwell, S.D.; Lucas, H.L. On territorial behaviour and other factor influencing habitat distribution in birds. Acta Biotheor.
**1969**, 19, 16–36. [Google Scholar] [CrossRef] - Lessells, C.M. Putting resource dynamics into continuous free distribution models. Anim. Behav.
**1995**, 49, 487–494. [Google Scholar] [CrossRef] - Křivan, V.; Cressman, R. The Ideal Free Distribution with travel costs. J. Theor. Biol.
**2024**, 579, 111717. [Google Scholar] [CrossRef] - Schwinning, S.; Rosenzweig, M.L. Periodic oscillation in an Ideal-Free predator-prey Distribution. OIKOS
**1990**, 59, 85–91. [Google Scholar] [CrossRef] - Bernstein, C.; Kacelnik, A.; Krebs, R. The Ideal Free Distribution and Predator-Prey Populations. TREE
**1992**, 7, 50–55. [Google Scholar] - Menezes, J.F.S.; Kotler, B.P. The generalized ideal free distribution model: Merging current ideal free distribution models into a central framework. Ecol. Model.
**2019**, 397, 47–54. [Google Scholar] [CrossRef] - Sirovnik, J.; Voelkl, B.; Keeling, L.J.; Würbel, H.; Toscano, M.J. Breakdown of the ideal free distribution under conditions of severe and low competition. Behav. Ecol. Sociobiol.
**2021**, 75, 31. [Google Scholar] [CrossRef] - Cressman, R.; Křivan, V. Migration Dynamics for the Ideal Free Distribution. Am. Nat.
**2006**, 168, 384–394. [Google Scholar] [CrossRef] - Auger, P.; Bernstein, C.; Poggiale, J.C. Predator Migration Decisions, the Ideal Free Distribution, and Predator-Prey Dynamics. Am. Nat.
**1999**, 153, 267–281. [Google Scholar] - Cressman, R.; Garay, G.; Křivan, V. Ideal Free Distributions, Evolutionary Games, and Population Dynamics in Multiple-Species Environments. Am. Nat.
**2004**, 164, 437–489. [Google Scholar] [CrossRef] - Bell, A.V.; Rader, R.B.; Peck, S.L.; Sih, A. The positive effects of negative interactions: Can avoidance of competitors or predators increase resource sampling by prey? Theor. Popul. Biol.
**2009**, 76, 52–58. [Google Scholar] [CrossRef] [PubMed] - Cantrell, R.S.; Cosner, C.; Lou, Y. Evolution of dispersal and the ideal free distribution. Math. Biosci. Eng.
**2010**, 7, 17–36. [Google Scholar] - Cantrell, R.S.; Cosner, C.; Lou, Y. Evolutionary stability of ideal free dispersal strategies in patchy environments. J. Math. Biol.
**2012**, 65, 943–965. [Google Scholar] [CrossRef] - Korobenko, L.; Braverman, E. On evolutionary stability of carrying capacity driven dispersal in competition with regularly diffusing populations. J. Math. Biol.
**2014**, 69, 1181–1206. [Google Scholar] [CrossRef] [PubMed] - Braverman, E.; Kamrujjaman, M. Competitive—Cooperative models with various diffusion strategies. Comput. Math. Appl.
**2016**, 72, 653–662. [Google Scholar] [CrossRef] - Cantrell, R.S.; Cosner, C.; Lam, K.-Y. Ideal free dispersal under general spatial heterogeneity and time periodicity. SIAM J. Appl. Math.
**2021**, 81, 789–813. [Google Scholar] [CrossRef] - Cantrell, R.S.; Cosner, C.; Zhou, Y. Ideal free dispersal in integrodifference models. J. Math. Biol.
**2022**, 85, 5. [Google Scholar] [CrossRef] - Epifanov, A.V.; Tsybulin, V.G. Mathematical Model of the Ideal Distribution of Related Species in a Nonhogeneous Environment. Vladikavkaz Math. J.
**2023**, 25, 78–88. (In Russian) [Google Scholar] - Zelenchuk, P.A.; Tsybulin, V.G. The ideal free distribution in a predator–prey model with multifactor taxis. Biophysics
**2021**, 66, 464–471. [Google Scholar] [CrossRef] - Holling, C.S. Some characteristics of simple types of predation and parasitism. Can. Entomol.
**1959**, 91, 385–398. [Google Scholar] [CrossRef] - Tyutyunov, Y.V.; Titova, L.I. Ratio-Dependence in Predator-Prey Systems as an Edge and Basic Minimal Model of Predator Interference. Can. Front. Ecol. Evol.
**2021**, 9, 725041. [Google Scholar] [CrossRef] - Ha, T.D.; Tsybulin, V.G.; Zelenchuk, P.A. How to model the local interaction in the predator–prey system at slow diffusion in a heterogeneous environment? Ecol. Complex.
**2022**, 52, 101026. [Google Scholar] [CrossRef]

**Figure 2.**Comparison of distributions of resource (green), prey (blue) and predator (red) truncated by Planes 1 and 4 (see Figure 1), ($\nu =1$): $n=1$, $m=1$, ${k}_{1}={k}_{2}=0.1$, ${\alpha}_{1}=0.2$, ${\alpha}_{3}=0.1$, ${\alpha}_{2}=0.1$, ${\mu}_{1}=1.3$, ${\lambda}_{1}=0.5$, ${\mu}_{2}=1.0$, ${\lambda}_{2}=0.7$.

**Figure 3.**Oscillations in the distributions of prey (blue) and predator (red) in case of violation of the stability criterion, when the area is cut by two planes (1 and 4): for a point in the area with coordinates $x=3$, $y=2$.

**Figure 5.**Comparison of prey (blue) and predator (red) distributions: $\nu =0$, $n=1$, $m=2$, ${k}_{1}={k}_{2}=0.1$, ${\alpha}_{1}=0.1$, ${\alpha}_{3}=0$, ${\alpha}_{2}=0.1$, ${\mu}_{1}=1.7$, ${\lambda}_{1}=0.4$, ${\mu}_{2}=1.0$, ${\lambda}_{2}=0.7$.

**Figure 6.**Comparison of resource (green), prey (blue) and predator (red) truncated by the planes (1 and 4): $n=1$, $m=3$, ${k}_{1}=0.2$, ${k}_{2}=0.1$, ${\alpha}_{1}=0.1$, ${\alpha}_{2}=0.1$, ${\alpha}_{3}=-0.1$, ${\mu}_{1}=1.5$, ${\lambda}_{1}=0.5$, ${\mu}_{2}=1.0$, ${\lambda}_{2}=0.7$.

**Figure 7.**Violation of IFD. Prey (blue) and predator (red) distributions in Section (Plane) 4 for different values $\nu $: IFD (solid lines) and deviation due to variation of diffusion parameter ${\widehat{k}}_{1}={k}_{1}+\epsilon $ (dashed lines); function resource (green); ${k}_{1}=0.2$, ${k}_{2}=0.1$, ${\alpha}_{1}=0.1$, ${\alpha}_{2}=0.1$, ${\alpha}_{3}$ = −0.1, ${\mu}_{1}=1.5$, ${\lambda}_{1}=0.5$, ${\mu}_{2}=1.0$, ${\lambda}_{2}=0.7$.

**Figure 8.**Violation of IFD. Prey (blue) and predator (red) distributions in Section (Plane) 4 for different values $\nu $: IFD (solid lines) and deviation due to variation of diffusion parameter ${\widehat{k}}_{2}={k}_{2}-\epsilon $ (dashed lines); function resource (green); ${k}_{1}=0.2$, ${k}_{2}=0.1$, ${\alpha}_{1}=0.1$, ${\alpha}_{2}=0.1$, ${\alpha}_{3}$ = −0.1, ${\mu}_{1}=1.5$, ${\lambda}_{1}=0.5$, ${\mu}_{2}=1.0$, ${\lambda}_{2}=0.7$.

$\mathit{\nu}$ | Conditions | |
---|---|---|

2 | ${\alpha}_{1}={k}_{1}+2{\alpha}_{2}$ | ${\alpha}_{3}=2{k}_{2}$ |

1 | ${\alpha}_{1}={k}_{1}+{\alpha}_{2}$ | ${\alpha}_{3}={k}_{2}$ |

0 | ${\alpha}_{1}={k}_{1}$ | ${\alpha}_{3}=0$ (any ${k}_{2}$) |

−1 | ${\alpha}_{1}={k}_{1}-{\alpha}_{2}$ | ${\alpha}_{3}=-{k}_{2}$ |

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**MDPI and ACS Style**

Tsybulin, V.; Zelenchuk, P.
Predator–Prey Dynamics and Ideal Free Distribution in a Heterogeneous Environment. *Mathematics* **2024**, *12*, 275.
https://doi.org/10.3390/math12020275

**AMA Style**

Tsybulin V, Zelenchuk P.
Predator–Prey Dynamics and Ideal Free Distribution in a Heterogeneous Environment. *Mathematics*. 2024; 12(2):275.
https://doi.org/10.3390/math12020275

**Chicago/Turabian Style**

Tsybulin, Vyacheslav, and Pavel Zelenchuk.
2024. "Predator–Prey Dynamics and Ideal Free Distribution in a Heterogeneous Environment" *Mathematics* 12, no. 2: 275.
https://doi.org/10.3390/math12020275