[Paper Review] Predators-Prey models with competition Part.1 : Existence, Bifurcation and qualitative properties

Henri Berestycki, Alessandro Zilio|arXiv (Cornell University)|Dec 5, 2017

Mathematical and Theoretical Epidemiology and Ecology Models18 references16 citations

TL;DR

This paper studies a reaction-diffusion model of predator-prey systems with inter-predator competition, proving existence, uniqueness, and asymptotic behavior of solutions. It shows that strong competition limits the number of coexisting predator groups, and surprisingly, optimal total predator populations emerge not from uniform groups but from multiple competing, aggressive predator subgroups.

ABSTRACT

International audience

Motivation & Objective

• To understand how self-interested, non-coordinated predator behavior can lead to territoriality in ecological systems.
• To analyze the existence, uniqueness, and long-term behavior of solutions in a reaction-diffusion model with competing predators and preys.
• To determine the conditions under which multiple predator groups can coexist and the maximum total predator population is achieved.
• To investigate the role of competition strength (β) in shaping spatial segregation and population distribution.
• To explore whether optimal predator population size is achieved through homogeneous groups or through competing, aggressive subgroups.

Proposed method

• Formulates a system of parabolic partial differential equations modeling prey (u) and N predator groups (w1,…,wN) with Lotka-Volterra-type dynamics and inter-group competition.
• Uses bifurcation analysis to show the existence of a rich set of non-constant stationary solutions, especially under strong competition (large β).
• Applies uniform estimates and asymptotic analysis to describe solutions in the limit β → +∞, showing convergence to segregated configurations.
• Employs comparison principles and eigenvalue estimates to analyze the behavior of solutions under extreme competition and large predation rates.
• Uses variational methods and optimization techniques to identify solutions that maximize the total predator population ∫Ω∑wi.
• Applies energy estimates and weak/strong convergence arguments to prove the vanishing of predator densities in the limit of high competition or large predation rates.

Experimental results

Research questions

• RQ1 Under what conditions do non-constant stationary solutions exist in a predator-prey system with competing predator groups?
• RQ2 How does increasing the competition strength β affect the number of coexisting predator groups and their spatial distribution?
• RQ3 Can the total predator population be maximized by a configuration with multiple competing predator subgroups rather than a single homogeneous group?
• RQ4 What is the upper bound on the number of coexisting predator groups in a given domain, and how does it depend on ecological parameters?
• RQ5 Does strong competition between predators lead to spatial segregation, and can this be described precisely in the limit β → +∞?

Key findings

• For sufficiently large competition strength β > β̄, solutions to the system are either close to constant (and small) or have at most N̄ + 1 non-trivial components, where N̄ ≲ |Ω|/(4π) max_i (λki − μωi)/diμ in two dimensions.
• In the limit β → +∞, solutions converge to segregated configurations where predator groups occupy disjoint territories, and limit solutions can be rigorously defined.
• Aggressive competition (high β) can prevent invasion by foreign predator groups, enhancing local stability.
• The total predator population ∫Ω∑wi is maximized not by a single homogeneous predator group, but by configurations with two or more competing predator subgroups, especially when the domain is rectangular and μ is small.
• Under strong competition and large predation rates, predator densities wn and prey density un converge uniformly to zero or to constant states, depending on parameter regimes.
• The analysis proves that optimal predator population configurations under competition require multiple, mutually aggressive predator groups, demonstrating that competition can be beneficial for total predator abundance.