On the effects of protection zone and directed population flux in prey-predator dynamics

Technical Summary: On the effects of protection zone and directed population flux in prey-predator dynamics

Problem Statement
This paper investigates a spatial predator-prey model incorporating two distinct ecological mechanisms: a protection zone (refuge) inaccessible to predators and a directed population flux where predators move toward prey-rich regions. The model is defined on a bounded domain $\Omega \subset \mathbb{R}^N$, with a subdomain $\Omega_0 \subset \Omega$ designated as a protection zone for the prey. Prey ($u$) diffuse randomly throughout $\Omega$, while predators ($v$) are confined to $\Omega_1 = \Omega \setminus \Omega_0$ and exhibit directed movement modeled by a nonlinear diffusion term $\nabla \cdot [u^2 \nabla(v/u)]$. This term represents "far-sighted" movement, where individuals move based on prey density at the destination, contrasting with the more common "near-sighted" chemotaxis terms ($-\nabla \cdot (v\nabla u)$) based on local gradients. The system is governed by the following equations with Neumann boundary conditions:
$$\begin{cases} u_t = \Delta u + u(\lambda - u - b \mathbb{1}_{\Omega_1}v), & (x, t) \in \Omega \times (0, T), \\ v_t = \Delta v + \alpha \nabla \cdot \left[ u^2 \nabla \left( \frac{v}{u} \right) \right] + v(\mu + cu - v), & (x, t) \in \Omega_1 \times (0, T), \end{cases}$$
where $\lambda$ and $\mu$ are the intrinsic growth rates of prey and predators, respectively, and $\alpha \ge 0$ measures the strength of the directed flux.

Methodology
The authors employ a combination of nonlinear evolution equation theory, bifurcation analysis, and asymptotic analysis to study both the time-dependent and stationary problems.

1. Local-in-time Well-posedness: For the nonstationary problem (specifically in the case $N=2$), the authors reformulate the system as a quasilinear abstract evolution equation in the Banach space $L^2(\Omega) \times L^2(\Omega_1)$. They utilize the theory of analytic semigroups developed by Yagi to establish the existence and uniqueness of local-in-time solutions. A key technical challenge addressed is the discontinuity of the predation term across the interface $\partial \Omega_0$ and the potential loss of ellipticity in the predator equation if the prey density becomes complex-valued in the abstract setting. To handle this, a cut-off function is applied to the coefficient of the principal part of the predator operator, ensuring uniform ellipticity while preserving the biological reality of non-negative solutions.

2. Stationary Analysis and Bifurcation: For the steady-state problem, the authors analyze the global structure of positive solutions (coexistence states). They utilize the Crandall-Rabinowitz local bifurcation theorem to identify bifurcation points from semitrivial solutions (where one species is extinct). The direction of bifurcation (supercritical vs. subcritical) is determined by calculating the derivative of the bifurcation curve with respect to the bifurcation parameter $\lambda$. Global bifurcation branches are then extended using Rabinowitz's global bifurcation theorem and a priori estimates derived from maximum principles and elliptic regularity.

3. Asymptotic Analysis ($\alpha \to \infty$): To understand the behavior under strong directed movement, the authors perform a singular limit analysis as the flux parameter $\alpha$ tends to infinity. They derive a limiting nonlinear elliptic system satisfied by the rescaled prey density $w = \lim \alpha u$ and the predator density $v$. The global bifurcation structure of this limiting system is analyzed using the Lyapunov-Schmidt reduction method near the blow-up point $\lambda = 0$.

Key Contributions and Results

• Local Well-posedness: The paper establishes the existence of unique, non-negative, local-in-time solutions for the nonstationary system, overcoming the difficulties posed by the discontinuous refuge interface and the nonlinear directed flux term.
• Bifurcation from Semitrivial States:
  • For $\mu \ge 0$ (predator growth rate non-negative), positive solutions bifurcate from the predator-only state $(0, \mu)$ at $\lambda = \sigma_1(b\mu \mathbb{1}_{\Omega_1})$. The authors identify a critical threshold $\alpha^*$ such that the bifurcation is supercritical (stable branch) if $\alpha < \alpha^*$ and subcritical (unstable branch initially) if $\alpha > \alpha^*$.
  • For $\mu < 0$, bifurcation occurs from the prey-only state $(\lambda, 0)$ at $\lambda = |\mu|/c$, and this bifurcation is always supercritical regardless of $\alpha$.
• Multiplicity and Turning Points: A central finding is that the combination of the protection zone and strong directed flux ($\alpha > \alpha^*$) induces a turning-point structure (saddle-node bifurcation) in the global bifurcation branch. This leads to the multiplicity of coexistence steady states for certain parameter ranges. Specifically, for $\lambda$ slightly below the bifurcation threshold, two distinct positive steady states may exist.
• Asymptotic Behavior as $\alpha \to \infty$:
  • In regimes where $\lambda$ is sufficiently large, strong directed movement drives the predator to extinction, and the system converges to the prey-only state $(\lambda, 0)$.
  • However, in regimes where $\lambda$ is small (specifically $0 < \lambda \le \sigma_1(b\mu \mathbb{1}_{\Omega_1})$), the system exhibits alternative asymptotic behaviors. Besides converging to the prey-only state, the system may approach a state where the rescaled prey density $w = \alpha u$ and predator density $v$ converge to a positive solution of the limiting system.
  • The analysis of the limiting system reveals that as $\lambda \to 0^+$, the rescaled prey density $w$ blows up at a rate of $1/\lambda$, while the predator density $v$ tends to extinction.
• Synergistic Effect: The paper demonstrates that the protection zone and directed flux act synergistically. While previous studies on models without refuges suggested that directed flux has limited impact on the coexistence region, this study shows that the presence of a protection zone allows strong directed movement to significantly expand the parameter space for coexistence and induce complex bifurcation structures (turning points) that are not present in purely diffusive models or models without refuges.

Significance and Claims
The authors claim that their work clarifies the nontrivial interplay between spatial protection and predator movement behavior. They highlight that strong directed movement, often intuitively expected to benefit the predator, can paradoxically allow the prey to persist at lower intrinsic growth rates due to the emergence of multiple stable coexistence configurations. The paper provides a rigorous mathematical foundation for understanding how "far-sighted" movement rules, distinct from chemotaxis, interact with refuge structures to shape the global dynamics of predator-prey systems. The results suggest that the combined effect of these mechanisms can reverse bifurcation directions and create bistability, offering a more nuanced view of population persistence in heterogeneous environments.