[Paper Review] Persistence of species in a predator-prey system with climate change and either nonlocal or local dispersal

Wonhyung Choi, Thomas Giletti+1 more|arXiv (Cornell University)|May 4, 2021

Mathematical and Theoretical Epidemiology and Ecology Models41 references41 citations

TL;DR

The paper analyzes predator-prey dynamics under a shifting climate, proving existence of forced waves (front and mixed front-pulse) and characterizing long-time persistence or extinction of prey and predator when diffusion is nonlocal or local, depending on climate-change speed relative to forced-wave speeds.

ABSTRACT

We are concerned with the persistence of both predator and prey in a diffusive predator-prey system with a climate change effect, which is modeled by a spatial-temporal heterogeneity depending on a moving variable. Moreover, we consider both the cases of nonlocal and local dispersal. In both these situations, we first prove the existence of forced waves, which are positive stationary solutions in the moving frames of the climate change, of either front or pulse type. Then we address the persistence or extinction of the prey and the predator separately in various moving frames, and achieve a complete picture in the local diffusion case. We show that the survival of the species depends crucially on how the climate change speed compares with the minimal speed of some pulse type forced waves.

Motivation & Objective

• Motivate understanding of how a moving environmental heterogeneity due to climate change affects predator-prey persistence.
• Develop existence results for forced wave solutions in both nonlocal and local dispersal settings.
• Characterize long-time outcomes (persistence vs extinction) in relation to climate-change speed and forced-wave speeds.
• Provide a complete persistence/extinction picture in the local diffusion case and partial results in the nonlocal case.

Proposed method

• Model a diffusive predator-prey system with climate change as a moving heterogeneous environment (alpha depending on x-st).
• Study both nonlocal dispersal (convolution with J_i) and classical local diffusion (u_xx, v_xx).
• Define forced waves as positive stationary solutions in the moving frame with speed s (front, pulse, and mixed front-pulse types).
• Prove existence of front and mixed front-pulse forced waves under biologically relevant conditions (e.g., b>1, ab<1).
• Derive critical speeds s*, s**, s** etc. as minimal speeds of forced waves and relate them to persistence thresholds in large-time behavior.
• Apply Schauder fixed-point and upper-lower solution techniques to construct wave profiles.

Experimental results

Research questions

• RQ1 What are the conditions for existence of forced waves (front and mixed front-pulse) in the predator-prey system with climate shift?
• RQ2 How does the climate-change speed s compare with minimal forced-wave speeds to determine persistence or extinction of prey and predator in nonlocal and local diffusion settings?
• RQ3 What is the long-time behavior of solutions to the Cauchy problem under different s relative to s*, s*, s** and local/nonlocal dispersal?
• RQ4 How does the interdependence of prey and predator densities affect the spreading/persistence thresholds when the environment shifts?
• RQ5 To what extent can results for nonlocal dispersal be extended to local diffusion?

Key findings

• There exists a positive front-type forced wave solving the moving-frame system whenever b>1 and ab<1 (Theorem 2.1).
• A mixed front-pulse forced wave exists only when the climate-change speed s is above a critical value s*, determined by prey or predator parameters (Theorem 2.2).
• In the nonlocal setting, prey survival requires s ≤ s* and predator survival (when prey at maximal density) requires s ≤ s*, with distinct regimes depending on s relative to s* and s*.
• When prey outpaces predator (s* > s*), persistent coexistence occurs in moving frames with speeds between s and s**, and extinction beyond s or s* depending on the regime (Theorems 2.3 and 2.4).
• For local diffusion, the same persistence/extinction structure holds with explicit speeds s* = 2√(d1 r1) and s* = 2√(d2 r2(b−1)); Theorem 2.7 shows persistence in intermediate moving frames (0 < s < min{s*, s*}).
• Additional results (Theorems 2.5 and 2.6) provide persistence in intermediate frames when s is below certain thresholds s** and s** for nonlocal settings, with positive liminf for u or v in specified zones.