Parametric resonance, chaos and spatial structure in the Lotka-Volterra model

TL;DR

This work analyzes a Lotka–Volterra predator–prey system with seasonal forcing implemented as a time-periodic carrying capacity $K(t)=1/\left(\kappa_0+\kappa_1\cos\omega t\right)$ and introduces a homotopy parameter $\alpha\in[0,1]$ to connect a time-invariant equilibrium to the driven system. For $\alpha=1$, a Floquet analysis around the coexistence point yields parametric resonances and subharmonic/harmonic instabilities; as $\alpha$ decreases, these periodic orbits persist and undergo period-doubling bifurcations, culminating in chaotic dynamics near $\alpha\approx0.1$. Extending to diffusion, the authors show that spatial patterns only emerge when the mean-field dynamics are chaotic, with no characteristic wavelength and with coarsening behavior, suggesting a diffusion–chaos balance influences spatial resilience. Overall, seasonal forcing can induce rich temporal dynamics that, coupled with diffusion, generate complex spatiotemporal structures that may enhance ecosystem resilience under fluctuating resources.

Abstract

We investigate the Lotka-Volterra model for predator-prey competition with a finite carrying capacity that varies periodically in time, modeling seasonal variations in nutrients or food resources. In the absence of time variability, the ordinary differential equations have an equilibrium point that represents coexisting predators and prey. The time dependence removes this equilibrium solution, but the equilibrium point is restored by allowing the predation rate also to vary in time. This equilibrium can undergo a parametric resonance instability, leading to subharmonic and harmonic time-periodic behavior, which persists even when the predation rate is constant. We also find period-doubling bifurcations and chaotic dynamics. If we allow the population densities to vary in space as well as time, introducing diffusion into the model, we find that variations in space diffuse away when the underlying dynamics is periodic in time, but spatiotemporal structure persists when the underlying dynamics is chaotic. We interpret this as a competition between diffusion, which makes the population densities homogeneous in space, and chaos, where sensitive dependence on initial conditions leads to different locations in space following different trajectories in time. Patterns and spatial structure are known to enhance resilience in ecosystems, suggesting that chaotic time-dependent dynamics arising from seasonal variations in carrying capacity and leading to spatial structure, might also enhance resilience.

Figures (5)

• Figure 1: (a) Subharmonic resonance tongue in the ODEs (\ref{['eqs:original_model']}) with $\alpha=1$, $a^*=1$ and $\kappa_0=0.25$, with $n$ and $\kappa_1$ varying. The $+$ symbols indicate the presence of a period-two (subharmonic) periodic orbit on a grid of parameter values, and the solid line is the neutral stability curve from the Floquet analysis. (b) A period-two orbit with $\alpha=1$ and $(n,\kappa_1)=(0.5, 0.24)$. The two magenta $+$ symbols indicate integer multiples of the forcing period. (c) With $\alpha=0$, the red circles indicate chaotic dynamics, and the other symbols indicate periodic orbits: light gray $+$ is period 1, black $+$ is period 2, black, light gray and light red $\times$ are periods 4, 8 and higher, including periodic windows within the chaotic parameter regime. (d) Chaotic orbit at $(n,\kappa_1)=(0.7, 0.24)$. The magenta $+$ symbols, spread over a wide range, indicate integer multiples of the forcing period. Data for this and other figures is available in Swailem2025
• Figure 2: Value of $a$ at times that are an integer multiple of the period (the stroboscopic map), after a transient of 10000 periods for each parameter value in the ODEs (\ref{['eqs:original_model']}). Parameter values are $a^*=1$, $\kappa_0=0.25$ and $(n,\kappa_1)=(0.7,0.24)$. Period-one orbits are represented by a single point, and period-two orbits, found below the period-doubling bifurcation ($\alpha\approx0.1076$), are represented by two points. Below $\alpha\approx0.1000$, the dynamics is chaotic.
• Figure 3: (a) The spatial fluctuations $a_{RMS}$ in the PDEs (\ref{['eqs:original_pde_model']}) decay exponentially with time for $\kappa_1=0.22$ and $0.23$, with $n=0.7$, so the flat state is stable in a domain of size $500\times500$. For $\kappa_1=0.24$, fluctuations grow and saturate: the flat state is unstable. (b) Phase portrait of the spatial averages $(\bar{a},\bar{b})$, shown as a black curve, with $(n,\kappa_1)=(0.7,0.24)$. The trajectory of $(\bar{a},\bar{b})$ in the PDEs is chaotic, but much less so that the light gray trajectory of $(a,b)$ from the ODEs at the same parameter values.
• Figure 4: Solution of the PDEs (\ref{['eqs:original_pde_model']}) showing the predator density $a(x,y)$ in a domain of size $500\times500$ with $\alpha=0$, $a^*=1$ and $\kappa_0=0.25$, after a transient of 10000 forcing periods. On each row, the left and right frames are one forcing period apart. (a,b) $(n,\kappa_1)=(0.6,0.24)$. (c,d) $(n,\kappa_1)=(0.7,0.24)$. In both cases, the ODEs are chaotic at these parameter values. The horizontal stripe in the top row is part of a very long transient. In both cases, there is no clear preferred length scale. A VisualPDE Walker2023 simulation of the PDEs is available on https://visualpde.com/sim/?mini=gmWq7tB4.
• Figure 5: A parameter survey confirms the link between chaos in the mean-field ODEs (\ref{['eqs:original_model']}) and the presence of spatial structure in the PDEs (\ref{['eqs:original_pde_model']}). The ODE data is the same as in, and the symbols have the same meaning as in, Fig. \ref{['fig:resonance_and_phase_portrait']}(c), though we have suppressed showing the period-one and period-two orbits. Red symbols indicate chaos in the ODEs, and the blue contour line separates PDE calculations with $a_{RMS}>10^{-5}$, so above the blue line, PDE solutions have persistent spatial structure.