Abstract
The interaction between a consumer (such as, a predator or a parasitoid) and a resource (such as, a prey or a host) forms an integral motif in ecological food webs, and has been modeled since the early 20th century starting from the seminal work of Lotka and Volterra. While the Lotka-Volterra predator-prey model predicts a neutrally stable equilibrium with oscillating population densities, a density-dependent predator attack rate is known to stabilize the equilibrium. Here, we consider a stochastic formulation of the Lotka-Volterra model where the prey’s reproduction rate is a random process, and the predator’s attack rate depends on both the prey and predator population densities. Analysis shows that increasing the sensitivity of the attack rate to the prey density attenuates the magnitude of stochastic fluctuations in the population densities. In contrast, these fluctuations vary non-monotonically with the sensitivity of the attack rate to the predator density with an optimal level of sensitivity minimizing the magnitude of fluctuations. Interestingly, our systematic study of the predator-prey correlations reveals distinct signatures depending on the form of the density-dependent attack rate. In summary, stochastic dynamics of nonlinear Lotka-Volterra models can be harnessed to infer density-dependent mechanisms regulating predator-prey interactions. Moreover, these mechanisms can have contrasting consequences on population density fluctuations, with predator-dependent attack rates amplifying stochasticity, while prey-dependent attack rates countering to buffer fluctuations.
Highlights
Predator-prey dynamics has been traditionally studied using an ordinary differential equation framework starting from the seminal work of Lotka and Volterra over a century ago [1–7]
We consider a stochastic formulation of the model by allowing the prey’s growth rate to follow an OrnsteinUhlenbeck random process that drives the deterministic predator-prey dynamics [3]. While both demographic and environmental stochasticity have been previously incorporated in predator-prey models, they have primarily focused on characterizing the role of noise in driving population extinctions and facilitating coexistence [44–47]. It remains to be seen how prey- and predator-dependent attack rates impact population density fluctuations, and to address this we take a novel moment-based approach using the Linear Noise Approximation technique to derive closed-form formulas quantifying the extent of fluctuations
A Type II functional response can provide stability in a narrow range if combined with other mechanisms, such as mutual interference between predators where fp < 0. Overall these results show that an attack rate that increases with prey density is sufficient to stabilize the equilibrium as long as −1 < fp < r fh
Summary
Predator-prey dynamics has been traditionally studied using an ordinary differential equation framework starting from the seminal work of Lotka and Volterra over a century ago [1–7]. A wide class of two-dimensional predator-prey models with an unstable equilibrium results in a stable limit cycle [29, 30] In this contribution, we focus on generalizing [1] to dhðtÞ 1⁄4 rhðtÞ À f ðh; pÞhðtÞpðtÞ ð3aÞ dpðtÞ 1⁄4 f ðh; pÞhðtÞpðtÞ À gpðtÞ ð3bÞ dt that considers a density-dependent attack rate f(h, p), where f is a continuously differentiable function in both arguments. While both demographic and environmental stochasticity have been previously incorporated in predator-prey models, they have primarily focused on characterizing the role of noise in driving population extinctions and facilitating coexistence [44–47] It remains to be seen how prey- and predator-dependent attack rates impact population density fluctuations, and to address this we take a novel moment-based approach using the Linear Noise Approximation technique to derive closed-form formulas quantifying the extent of fluctuations. Our results show how simple statistical signatures (such as, the correlation between population densities) can inform density-dependent mechanisms at play in regulating population dynamics
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