[Paper Review] Bistability induced by generalist natural enemies can reverse pest invasions

Sten Madec, Jérôme Casas+2 more|arXiv (Cornell University)|Sep 7, 2015

Mathematical and Theoretical Epidemiology and Ecology ModelsMedicine43 references17 citations

TL;DR

This paper demonstrates that generalist natural enemies can reverse pest invasions in spatially explicit predator-prey systems with logistic prey growth, using reaction-diffusion models. Through analytical and numerical methods, it identifies that bistable dynamics—driven by predator searching efficiency and functional response—enable spatial control even without Allee effects, revealing new conditions for biological control beyond classical models.

ABSTRACT

Reaction-diffusion analytical modeling of predator-prey systems has shown that specialist natural enemies can slow, stop and even reverse pest invasions, assuming that the prey population displays a strong Allee effect in its growth. Few additional analytical results have been obtained for other spatially distributed predator-prey systems, as traveling waves of non-monotonous systems are notoriously difficult to obtain. Traveling waves have indeed recently been shown to exist in predator-prey systems, but the direction of the wave, an essential item of information in the context of the control of biological invasions, is generally unknown. Preliminary numerical explorations have hinted that control by generalist predators might be possible for prey populations displaying logistic growth. We aimed to formalize the conditions in which spatial biological control can be achieved by generalists, through an analytical approach based on reaction-diffusion equations. The population of the focal prey - the invader - is assumed to grow according to a logistic function. The predator has a type II functional response and is present everywhere in the domain, at its carrying capacity, on alternative hosts. Control, defined as the invader becoming extinct in the domain, may result from spatially independent demographic dynamics or from a spatial extinction wave. Using comparison principles, we obtain sufficient conditions for control and for invasion, based on scalar bistable partial differential equations (PDEs). The searching efficiency and functional response plateau of the predator are identified as the main parameters defining the parameter space for prey extinction and invasion. Numerical explorations are carried out in the region of those control parameters space between the super-and subso-lutions, in which no conclusion about controllability can be drawn on the basis of analytical solutions. The ability of generalist predators to control prey populations with logistic growth lies in the bis-table dynamics of the coupled system, rather than in the bistability of prey-only dynamics as observed for specialist predators attacking prey populations displaying Allee effects. The consideration of space in predator-prey systems involving generalist predators with a parabolic functional response is crucial. Analysis of the ordinary differential equations (ODEs) system identifies parameter regions with monostable (extinction) and bistable (extinction or invasion) dynamics. By contrast, analysis of the associated PDE system distinguishes different and additional regions of invasion and extinction. Depending on the relative positions of these different zones, four patterns of spatial dynamics can be identified : traveling waves of extinction and invasion, pulse waves of extinction and heterogeneous stationary positive solutions of the Turing type. As a consequence, prey control is predicted to be possible when space is considered in additional situations other than those identified without considering space. The reverse situation is also possible. None of these considerations apply to spatial predator-prey systems with specialist natural enemies.

Motivation & Objective

• To investigate whether generalist predators can reverse pest invasions in spatially distributed systems with logistic prey growth.
• To determine the conditions under which spatial biological control is possible, especially when prey do not exhibit Allee effects.
• To analyze the role of predator diffusion, functional response, and searching efficiency in determining invasion or extinction outcomes.
• To identify the emergence of distinct spatial patterns—traveling waves, pulse waves, and Turing-type patterns—beyond classical monostable or bistable dynamics.
• To compare the dynamics of generalist versus specialist predator systems in spatial control contexts, highlighting the critical role of space in generalist systems.

Proposed method

• Formalizing the system using a reaction-diffusion model with logistic growth for prey and a type II functional response for generalist predators.
• Applying comparison principles to derive sufficient conditions for prey extinction and invasion using scalar bistable PDEs.
• Analyzing the associated ordinary differential equation (ODE) system to identify monostable (extinction) and bistable (extinction or invasion) parameter regions.
• Extending analysis to the PDE system to detect additional spatial dynamics, including traveling waves of extinction and invasion, pulse waves, and heterogeneous stationary solutions.
• Conducting numerical explorations in the parameter space between super- and subsolutions where analytical conclusions are inconclusive.
• Using mathematical tools such as traveling wave theory, stability analysis, and Turing instability detection to classify spatial patterns.

Experimental results

Research questions

• RQ1 Can generalist predators reverse pest invasions when prey grow logistically, without a strong Allee effect?
• RQ2 What are the key predator parameters—specifically searching efficiency and functional response plateau—that determine control or invasion outcomes?
• RQ3 How does spatial structure alter the dynamics of predator-prey systems with generalist predators compared to non-spatial or specialist predator systems?
• RQ4 What types of spatial patterns (e.g., traveling waves, pulse waves, Turing patterns) emerge in the PDE system, and what do they imply for biological control?
• RQ5 Why do generalist predators with higher diffusion coefficients fail to control prey, as suggested by prior simulations and field observations?

Key findings

• Generalist predators can reverse pest invasions in systems with logistic prey growth, provided the predator’s searching efficiency and functional response plateau are sufficiently high.
• The key mechanism enabling control is bistable dynamics in the coupled predator-prey system, not prey-only Allee effects as seen in specialist predator systems.
• Four distinct spatial dynamics emerge: traveling waves of extinction and invasion, pulse waves of extinction, and heterogeneous stationary solutions of the Turing type.
• The parameter space for control expands significantly when space is considered, revealing new conditions for extinction that do not exist in non-spatial models.
• Numerical explorations in the intermediate parameter region (between super- and subsolutions) confirm the existence of complex spatial patterns not predictable from ODE analysis alone.
• The study confirms that predator diffusion rate relative to prey diffusion is a critical factor, with higher predator diffusion reducing control efficacy—aligning with prior field and simulation observations.