[Paper Review] Stationary and time periodic patterns of two-predator and one-prey systems with prey-taxis

Ke Wang, Qi Wang+1 more|arXiv (Cornell University)|Aug 17, 2015

Mathematical Biology Tumor Growth46 references4 citations

TL;DR

This paper studies pattern formation in a three-species predator-prey system with two predators and one prey, incorporating prey-taxis—directed movement of predators toward higher prey density. It establishes global existence and boundedness of solutions, demonstrates that prey-taxis can stabilize or destabilize equilibria depending on group defense, and proves the existence of stationary and time-periodic nontrivial patterns via bifurcation analysis, with stability analysis revealing a wave mode selection mechanism.

ABSTRACT

This paper concerns pattern formation in a class of reaction-advection-diffusion systems modeling the population dynamics of two predators and one prey. We consider the biological situation that both predators forage along the population density gradient of the preys which can defend themselves as a group. We prove the global existence and uniform boundedness of positive classical solutions for the fully parabolic system over a bounded domain with space dimension $N=1,2$ and for the parabolic- -parabolic-elliptic system over higher space dimensions. Linearized stability analysis shows that prey-taxis stabilizes the positive constant equilibrium if there is no group defense while it destabilizes the equilibrium otherwise. Then we obtain stationary and time-periodic nontrivial solutions of the system that bifurcate from the positive constant equilibrium. Moreover, the stability of these solutions is also analyzed in detail which provides a wave mode selection mechanism of nontrivial patterns for this strongly coupled system. Finally, we perform numerical simulations to illustrate and support our theoretical results.

Motivation & Objective

• To understand how prey-taxis influences spatial pattern formation in a two-predator, one-prey ecological system.
• To investigate the impact of group defense in prey on the stability of predator-prey equilibria under prey-taxis.
• To establish the existence of nontrivial stationary and time-periodic solutions bifurcating from the positive constant equilibrium.
• To analyze the stability of these nontrivial patterns and identify a wave mode selection mechanism in the strongly coupled system.
• To validate theoretical findings through numerical simulations of the reaction–advection–diffusion system.

Proposed method

• Formulates a fully parabolic 3×3 reaction–advection–diffusion system modeling two predators and one prey with Lotka–Volterra kinetics and prey-taxis.
• Uses linearized stability analysis around the positive constant equilibrium to determine whether prey-taxis stabilizes or destabilizes the equilibrium, depending on the presence of group defense.
• Applies center manifold and Lyapunov–Schmidt reduction techniques to prove the existence of stationary and time-periodic nontrivial solutions via bifurcation theory.
• Derives explicit expressions for bifurcating solutions using Fourier expansion and solves associated matrix systems involving determinants of coefficient matrices.
• Performs numerical simulations to illustrate and support theoretical results on pattern formation and stability.
• Analyzes the stability of bifurcating solutions using the method of multiple scales and evaluates the second-order term in the expansion to determine wave mode selection.

Experimental results

Research questions

• RQ1 How does prey-taxis affect the stability of the positive constant equilibrium in a two-predator, one-prey system?
• RQ2 Under what conditions does prey-taxis lead to the emergence of nontrivial stationary or time-periodic patterns?
• RQ3 How does group defense in prey influence the role of prey-taxis in pattern formation and equilibrium stability?
• RQ4 What mechanism governs the selection of wave modes in the emerging nontrivial patterns?
• RQ5 What are the conditions under which global existence and uniform boundedness of classical solutions hold in the system?

Key findings

• Global existence and uniform boundedness of positive classical solutions are proven for the fully parabolic system in space dimensions N = 1, 2, and for the parabolic–parabolic–elliptic system in higher dimensions.
• Prey-taxis stabilizes the positive constant equilibrium when there is no group defense in prey, but destabilizes it when prey exhibit group defense.
• Nontrivial stationary and time-periodic solutions bifurcate from the positive constant equilibrium under appropriate parameter conditions.
• The stability of these bifurcating solutions is analyzed in detail, revealing a wave mode selection mechanism based on the second-order term in the bifurcation expansion.
• Numerical simulations confirm the theoretical predictions, showing aggregation of predators in high-prey-density regions and the emergence of complex spatiotemporal patterns.
• Explicit expressions for the amplitudes of bifurcating solutions are derived using determinants of coefficient matrices from Fourier expansion and integration by parts.