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Modeling Predator-Prey Dynamics with Stochastic Differential Equations: Patterns of Collective Hunting and Nonlinear Predation Effects

Junyi Qi, Ton Viet Ta

TL;DR

We address how predator group size and hunting strategies shape prey survival in aquatic schooling systems by developing an agent-based framework of coupled stochastic differential equations (SDEs) that track $N$ prey and $M$ predators in $\mathbb{R}^d$. The model encodes attraction, repulsion, alignment, and stochastic noise, and implements two hunting strategies—center attack and nearest attack—through a force $F(\mathbf{X},\mathbf{V},\mathbf{Y},\mathbf{U})$ and a capture radius $d_{\text{capture}}$. Key findings reveal nonlinear benefits of cooperative predation: predation efficiency rises with predator number at first but declines beyond a critical size due to intra-predator interference, producing diverse emergent patterns such as dispersal, fragmentation, oscillatory predation, and encirclement. The framework provides a generalizable tool for studying predator–prey interactions, with implications for ecological stability, conservation, and bio-inspired computation, and can be extended to multi-species and heterogeneous environments.

Abstract

We investigate predator-prey school interactions in aquatic environments using a stochastic differential equation (SDE)-based, particle-level model that incorporates attraction, repulsion, alignment, and environmental noise. Two predation strategies-center attack and nearest attack-are examined to assess their effects on prey survival, predator efficiency, and group dynamics. Simulations reveal diverse emergent behaviors such as prey dispersal and regrouping, oscillatory predation with collective defense, and predator encirclement. Results show that collective hunting enhances capture efficiency compared to solitary attacks, but benefits diminish beyond a critical predator group size due to intra-predator competition. This work provides new insights into cooperative predation and introduces a generalizable SDE framework for analyzing predator-prey interactions.

Modeling Predator-Prey Dynamics with Stochastic Differential Equations: Patterns of Collective Hunting and Nonlinear Predation Effects

TL;DR

We address how predator group size and hunting strategies shape prey survival in aquatic schooling systems by developing an agent-based framework of coupled stochastic differential equations (SDEs) that track prey and predators in . The model encodes attraction, repulsion, alignment, and stochastic noise, and implements two hunting strategies—center attack and nearest attack—through a force and a capture radius . Key findings reveal nonlinear benefits of cooperative predation: predation efficiency rises with predator number at first but declines beyond a critical size due to intra-predator interference, producing diverse emergent patterns such as dispersal, fragmentation, oscillatory predation, and encirclement. The framework provides a generalizable tool for studying predator–prey interactions, with implications for ecological stability, conservation, and bio-inspired computation, and can be extended to multi-species and heterogeneous environments.

Abstract

We investigate predator-prey school interactions in aquatic environments using a stochastic differential equation (SDE)-based, particle-level model that incorporates attraction, repulsion, alignment, and environmental noise. Two predation strategies-center attack and nearest attack-are examined to assess their effects on prey survival, predator efficiency, and group dynamics. Simulations reveal diverse emergent behaviors such as prey dispersal and regrouping, oscillatory predation with collective defense, and predator encirclement. Results show that collective hunting enhances capture efficiency compared to solitary attacks, but benefits diminish beyond a critical predator group size due to intra-predator competition. This work provides new insights into cooperative predation and introduces a generalizable SDE framework for analyzing predator-prey interactions.

Paper Structure

This paper contains 14 sections, 1 theorem, 14 equations, 20 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Model Description
  3. Prey school dynamics
  4. Predator school dynamics
  5. Regularization via $\epsilon$
  6. Stochastic effects
  7. Hunting strategies
  8. Initial conditions
  9. Eaten condition
  10. Predation Patterns
  11. Impact of predator school size on prey dynamics
  12. Nearest Attack Strategy
  13. Center Attack Strategy
  14. Conclusion

Key Result

Theorem 2.1

Let the initial state be Then the system eq:dynamic_predation_system admits a unique local solution on some interval $[0,\tau)$, where $\tau \leq \infty$. If $\tau < \infty$ almost surely, then $\tau$ is an explosion time.

Figures (20)

  • Figure 1: Graphs of $f_1(x)$ and $f_2(x)$ in \ref{['two_forces']}, with parameters $p=2$, $q=3$, and $\epsilon=0.01.$
  • Figure 3: Pattern 2: Dispersal by predators under the center attack strategy. Predators (black dots) disrupt the cohesion of the prey school (blue dots), forcing individuals to scatter. Time steps (left to right): $t = 0$, $650$, $1100$, $2999$.
  • Figure 4: Pattern 3: Selective pursuit after dispersal under the nearest attack strategy. After dispersing the prey school (blue dots), predators (black dots) pursue the nearest prey individually, resulting in decentralized hunting. Time steps (left to right, top to bottom): $t = 0$, $100$, $240$, $370$, $500$, $600$, $700$, $920$.
  • Figure 5: Pattern 4: Fragmentation and regrouping. Predators (black dots) fragment the prey school (blue dots), then engage in dispersed hunting before regrouping to target a single cluster. Time steps (left to right, top to bottom): $t = 0$, $300$, $600$, $740$, $840$, $1500$, $1720$, $2999$.
  • Figure 6: Pattern 5: Edge dispersal and selfish escape under the nearest attack strategy. Predators (black dots) exert pressure on edge prey, causing individuals to peel away from the school (blue dots) in selfish escape attempts. Time steps (left to right, top to bottom): $t = 0$, $200$, $500$, $580$, $660$, $780$, $980$, $1400$.
  • ...and 15 more figures

Theorems & Definitions (2)

  • Theorem 2.1
  • proof