[Paper Review] Multiple Patterns Formation for an Aggregation/Diffusion Predator-Prey System

Simone Fagioli, Yahya Jaafra|arXiv (Cornell University)|Apr 10, 2019

Mathematical and Theoretical Epidemiology and Ecology ModelsMedicine37 references3 citations

TL;DR

This paper studies a one-dimensional aggregation/diffusion predator-prey system with non-local interactions and quadratic diffusion, proving the existence of stationary solutions composed of multiple spatial patterns (bumps) via the functional Implicit Function Theorem for small diffusion. The key result shows that under specific conditions on the interaction strength $\alpha$ and bump positions, multiple-bump steady states emerge, validated by numerical simulations demonstrating mixed, separated, and traveling wave patterns.

ABSTRACT

We investigate existence of stationary solutions to an aggregation/diffusion system of PDEs, modelling a two species predator-prey interaction. In the model this interaction is described by non-local potentials that are mutually proportional by a negative constant $-α$, with $α>0$. Each species is also subject to non-local self-attraction forces together with quadratic diffusion effects. The competition between the aforementioned mechanisms produce a rich asymptotic behaviour, namely the formation of steady states that are composed of multiple bumps, i.e. sums of Barenblatt-type profiles. The existence of such stationary states, under some conditions on the positions of the bumps and the proportionality constant $α$, is showed for small diffusion, by using the functional version of the Implicit Function Theorem. We complement our results with some numerical simulations, that suggest a large variety in the possible strategies the two species use in order to interact each other.

Motivation & Objective

• To investigate the existence of stationary solutions in a non-local aggregation/diffusion system modeling predator-prey dynamics with mutual non-local attraction and repulsion.
• To analyze the formation of complex spatial patterns—specifically multiple-bump structures—arising from the competition between self-attraction, cross-attraction/repulsion, and quadratic diffusion.
• To establish conditions on the interaction strength $\alpha$ and spatial configuration of bumps under which such multi-bump steady states exist.
• To validate theoretical findings through numerical simulations showing diverse dynamic behaviors including mixed, separated, and traveling wave patterns.

Proposed method

• Formal derivation of the PDE system from a particle-based model with non-local self-interaction and predator-prey interaction potentials.
• Reduction to a one-dimensional system with symmetric diffusion coefficients via scaling, simplifying analysis while preserving key dynamics.
• Application of the functional version of the Implicit Function Theorem to prove existence of multi-bump stationary solutions for small diffusion.
• Use of Barenblatt-type profiles as base solutions to construct multi-bump steady states through perturbation analysis.
• Numerical simulations using both particle methods and finite volume schemes to visualize and validate the emergence of complex patterns.
• Employment of normalized Gaussian kernels for all interaction potentials ($S_\rho, S_\eta, K$) to ensure smoothness and satisfy analytical assumptions.

Experimental results

Research questions

• RQ1 Under what conditions on the interaction strength $\alpha$ and spatial configuration can multiple-bump stationary solutions exist in the aggregation/diffusion predator-prey system?
• RQ2 How does the competition between self-attraction, cross-attraction/repulsion, and quadratic diffusion lead to the formation of stable spatial patterns?
• RQ3 Can the theoretical existence of multi-bump steady states be numerically validated across varying initial data and parameter regimes?
• RQ4 What dynamic behaviors—such as mixed states, separated configurations, or traveling waves—emerge from different initial conditions and values of $\alpha$?
• RQ5 How does the value of $\alpha$ influence the transition between coexistence (mixed) and segregation (separated) steady states?

Key findings

• For small diffusion and specific configurations of bump positions, the paper proves the existence of stationary solutions composed of multiple bumps via the Implicit Function Theorem.
• Numerical simulations confirm the existence of a four-bump steady state with $\alpha = 0.05$, $d = 0.3$, and initial data inducing balanced attractive forces.
• A five-bump steady state is numerically observed with $\alpha = 1$, $d = 0.3$, and initial data placing predators and prey in symmetric clusters.
• With $\alpha = 6$, a transition from a mixed to a separated steady state is observed, indicating high escape propensity of prey from predators.
• A traveling wave pattern emerges with $\alpha = 1$, $d = 0.2$, and initial data placing predators and prey in spatially separated regions, suggesting dynamic coexistence.
• The choice of $\alpha$ critically determines system behavior: low $\alpha$ leads to mixed states, high $\alpha$ to separation, and intermediate values to traveling waves.