Spatio-temporal pattern formation under varying functional response parametrizations

TL;DR

The paper investigates how spatial pattern formation in a two-species Bazykin predator–prey system depends on the choice of prey-dependent functional response, extending structural sensitivity analysis from nonspatial ODEs to spatio-temporal PDEs with diffusion. By developing a generalized functional response $f(u)$ and analyzing both temporal and diffusive systems, the authors derive the full equilibrium and bifurcation structure (Saddle-Node, Cusp, Hopf, Bogdanov–Takens) and establish conditions for Turing instabilities, as well as existence, nonexistence, and bounds of heterogeneous steady states. They validate the theory with numerical simulations for Holling Type II and Ivlev responses, showing that switching between parametrizations shifts bifurcation thresholds and yields markedly different pattern types, including stationary hotspots and labyrinthines, and can even produce patterns outside traditional Turing domains. The results highlight the sensitivity of spatio-temporal patterns to how ecological interactions are mathematically represented, underscoring the need to account for structural sensitivity when forecasting species persistence and pattern formation in spatially explicit ecosystems.

Abstract

Enhancement of the predictive power and robustness of nonlinear population dynamics models allows ecologists to make more reliable forecasts about species' long term survival. However, the limited availability of detailed ecological data, especially for complex ecological interactions creates uncertainty in model predictions, often requiring adjustments to the mathematical formulation of these interactions. Modifying the mathematical representation of components responsible for complex behaviors, such as predation, can further contribute to this uncertainty, a phenomenon known as structural sensitivity. Structural sensitivity has been explored primarily in non-spatial systems governed by ordinary differential equations (ODEs), and in a limited number of simple, spatially extended systems modeled by nonhomogeneous parabolic partial differential equations (PDEs), where self-diffusion alone cannot produce spatial patterns. In this study, we broaden the scope of structural sensitivity analysis to include spatio-temporal ecological systems in which spatial patterns can emerge due to diffusive instability. Through a combination of analytical techniques and supporting numerical simulations, we show that pattern formation can be highly sensitive to how the system and its associated ecological interactions are mathematically parameterized. In fact, some patterns observed in one version of the model may completely disappear in another with a different parameterization, even though the underlying properties remain unchanged.

Figures (10)

• Figure 1: (a) 2d Bifurcation for Holling type II (b) 2d Bifurcation for Ivlev, with $X_0$ and $k_3$ as bifurcation parameters. Here $X_0$ is varying along $x$-axis and $k_3$ is varying along $y$-axis.
• Figure 2: Three parametric bifurcation diagram for the system \ref{['algeb_equn']} for the fixed parameter values $r=0.24$, $b_{H}=1.3$, $a_{H}=1$, $b_{I}=1.2367$, $a_{I}=0.4101$. The surface edged by the blue curves represents the Hopf bifurcation, the surface enclosed by the red and magenta lines denote two saddle-node bifurcation. The cyan curve representing cusp bifurcation, and the black curve indicates the Bogdanov–Takens bifurcation.
• Figure 3: One parametric bifurcation diagram for the system \ref{['algeb_equn']} with (a) the Holling type II functional response with parameter values corresponding to functional response are $k_{1_H}=1.3, k_{2_H}=1$, (b) the Ivlev functional response with parameter values corresponding to functional response are $k_{1_I}=1.2367, k_{2_I}=0.4101$. The fixed parameter values are $X_0=23~\hbox{&}~ r=0.24$. $k_3$ is the bifurcation parameter
• Figure 4: One parametric bifurcation diagram for the system \ref{['algeb_equn']} with (a) the Holling type II functional response with parameter values corresponding to functional response are $k_{1_H}=1.3, k_{2_H}=1$, (b) the Ivlev functional response with parameter values corresponding to functional response are $k_{1_I}=1.2367, k_{2_I}=0.4101$. The fixed parameter values are $X_0=52~\hbox{&}~ r=0.24$. $k_3$ is the bifurcation parameter
• Figure 5: (a) Three parametric plot of Turing and Hopf bifurcation curves for the system \ref{['alg_eqn_pde']} with the fixed parameter values $X_0=23$, $r=0.24$, $b_{H}=1.3$, $a_{H}=1$, $b_{I}=1.2367$, $a_{I}=0.4101$, representing a continuous shift of Turing bifurcation threshold due to continuous switch between two parameterizations of functional response: Holling's $\sigma=0$ to Ivlev's $\sigma=1$. (b) The plot of the Turing bifurcation curve (black) and Hopf bifurcation threshold (blue) in $(k_3,d)$-parameter space for the model with Holling type II functional response. (c) The plot of the Turing bifurcation curve (black) and Hopf bifurcation threshold (blue) in $(k_3,d)$-parameter space for the model with Ivlev functional response.

Theorems & Definitions (18)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof