Spreading speeds for prey--predator systems in a shifting environment: a short proof

Technical Summary: Spreading Speeds for Prey–Predator Systems in a Shifting Environment

Problem Statement
This paper investigates the spreading speed of the predator component in a class of prey–predator reaction–diffusion systems subject to a shifting environment. The system is modeled by the following equations:
$$\begin{cases} \partial_t u = d\partial_{xx}u + u f(x - c_1 t, u, v), \\ \partial_t v = \partial_{xx}v + v g(x - c_1 t, u, v), \end{cases}$$
where $u$ and $v$ represent prey and predator densities, respectively, $d$ is the prey diffusion rate, and $c_1 > 0$ is the velocity of the shifting habitat. The interaction functions $f$ and $g$ depend on the moving variable $z = x - c_1 t$, representing spatiotemporal heterogeneity. The primary goal is to determine the rightward spreading speed of the predator ($v$) when the prey is initially uniformly established.

A central difficulty addressed is that the full system lacks a direct comparison principle due to the non-cooperative nature of the interaction (predation reduces prey growth, while prey density drives predator growth). Furthermore, the environmental profile $g(z, 1, 0)$ may be nonmonotone, a scenario common in biological modeling where growth rates depend unimodally on temperature or other factors.

Methodology
The author employs a reduction strategy to bypass the lack of a comparison principle for the coupled system. The core methodological contribution is the establishment of a pointwise estimate (Lemma 4.1), which demonstrates that in regions where the predator density is sufficiently small, the prey density is close to its carrying capacity (normalized to 1).

Specifically, the proof proceeds as follows:

1. Invariant Region and Bounds: Using scalar comparison principles for each component separately, the author establishes that solutions remain within a bounded invariant region $[0, 1] \times [0, V]$.
2. Pointwise Estimate: By contradiction and the use of parabolic estimates (blow-up arguments), it is shown that for any $\alpha > 0$, there exist constants such that $1 - u(t, x) \leq \alpha + M_\alpha v(t, x)$ for large $t$. This implies that where $v$ is small, $u \approx 1$.
3. Reduction to Scalar KPP: This estimate allows the predator equation to be bounded below by a scalar Fisher–KPP equation with an effective growth rate $g(x - c_1 t, 1, 0)$. The upper bound is similarly derived using the monotonicity of $g$ with respect to $v$.
4. Eigenvalue Analysis: The spreading speed is determined by analyzing the generalized principal eigenvalue $\Lambda_1$ of the linearized operator associated with the effective growth rate $g(z, 1, 0)$ in the shifting frame.

Key Results
The main theorem (Theorem 2.7) provides an explicit formula for the spreading speed $c^*_v$ of the predator, which depends on the shifting speed $c_1$, the asymptotic growth rates $r_\pm = \lim_{z\to\pm\infty} g(z, 1, 0)$, and the generalized principal eigenvalue $\Lambda_1$. The speed is given by a piecewise function:

• Case 1 ($0 < c_1 \leq 2\sqrt{r_+}$): $c^*_v = 2\sqrt{r_+}$. The speed is linearly determined by the leading edge growth rate.
• Case 2 ($2\sqrt{r_+} \leq c_1 \leq 2\sqrt{\Lambda_1}$): $c^*_v = c_1$. This is the locking phenomenon, where the predator propagates at the same speed as the shifting environment.
• Case 3 ($2\sqrt{\Lambda_1} \leq c_1 \leq 2(\sqrt{r_-} + \sqrt{\Lambda_1 - r_-})$): The speed is given by a complex formula involving $c_1, \Lambda_1,$ and $r_-$. This corresponds to the nonlocal pulling phenomenon, where distant favorable habitats enhance the spreading speed beyond what the local leading edge would suggest.
• Case 4 ($c_1 \geq 2(\sqrt{r_-} + \sqrt{\Lambda_1 - r_-})$): $c^*_v = 2\sqrt{r_-}$.

The paper also establishes asymptotic behaviors (Corollaries 2.10 and 2.13). It shows that ahead of the front, the prey approaches its carrying capacity and the predator vanishes. Behind the front, the solution converges to a coexistence equilibrium determined by the limiting environment ($f_\pm, g_\pm$) seen in the moving frame, provided the limiting ODE systems admit strict Lyapunov functions.

Significance and Claims
The paper claims to provide a rigorous reduction from a non-cooperative prey–predator system to an associated scalar Fisher–KPP equation in a shifting environment, even when the environmental profile is nonmonotone.

• Extension of Scalar Theory: The results confirm that phenomena previously discovered for scalar KPP equations—specifically locking and nonlocal pulling—persist at the level of predator invasion in non-cooperative systems.
• Handling Nonmonotonicity: The methodology successfully addresses the difficulties introduced by nonmonotone shifting environments, which are biologically significant but mathematically challenging.
• Conjecture Confirmation: The main result confirms a conjecture regarding the locking case stated in prior work [16, Remark 3], which had used a Hamilton–Jacobi approach for a specific monotone case. This paper extends the validity of the locking phenomenon to nonmonotone settings.
• Limitations: The author notes that while the spreading speed is characterized, the logarithmic delay correction (Bramson's correction) for the position of the solution front remains an open problem for prey–predator systems, and the current argument does not address this.

The paper concludes by applying these general results to a Lotka–Volterra model with a shifting conversion rate, demonstrating the explicit calculation of spreading speeds and the verification of asymptotic convergence to coexistence states.