Rosenzweig-MacArthur Model Research Articles

How localized consumption stabilizes predator-prey systems with finite frequency of mixing.

Predator-prey theory began with aspatial models that assumed organisms interacted as if they were "well-mixed" particles that obey the laws of mass action, but it has become clear that both the spatial and individual nature of many organisms can change how the dynamics of such systems function. Here I examine how localized consumption of prey by predators changes the dynamics of predator-prey systems; I use an individual-based simulation of the Rosenzweig-MacArthur model in implicit space and its mean-field approximation. In combination with limited movement, localized consumption makes the predator-prey dynamics more stable than the comparable "well-mixed" Rosenzweig-MacArthur model. Using a spatial correlation, one can directly compare a simplified version of the individual-based model with the Rosenzweig-MacArthur model. While this comparison allows the changes in the dynamics to be captured by the "well-mixed" Rosenzweig-MacArthur model, the parameters of the functional response are now dependent on the movement parameters, and so the functional response must be estimated statistically from the dynamics of the individual-based model. Yet this implies that aspatial models may work in a scale-specific fashion for spatial systems. Unlike many recent spatial models, the localized consumption and limited movement in the model presented here cannot produce coherent spatial patterns and do not depend on a patchy structure, as found in metapopulation models. Instead, the individual nature of the interactions creates a diffusion-limited reaction, which appears closer to a form of ephemeral refuge.

THRESHOLD POLICIES IN THE CONTROL OF PREDATOR-PREY MODELS

Threshold policies are defined and analysed for different types of one species and predator-prey type models. It is shown that such policies can be designed by suitable choice of so called virtual equilibrium points. The simplest threshold policies are discontinuous, which leads to some drawbacks. It is also shown how to design continuous threshold policies that retain most of the advantages of their discontinuous versions but do not have the major drawback of chatter in the control. Threshold policies are also seen to be robust to uncertainty of model parameters, and initial conditions, as well as to delays between stock assessment and policy enactment. The models which are controlled are the Noy-Meir herbivore-vegetation model, subject to linear consumption curves, the two species Rosenzweig-MacArthur model, and a three-dimensional chemostat model.

Predator migration in response to prey density: what are the consequences?

A predator-prey metapopulation model with two identical patches and only migration of the predator is investigated. Local predator-prey interaction is described by the so-called Rosenzweig-MacArthur model, while the migration term of the predator is put in a nonlinear form, which is derived by extending the Holling time budget argument to migration. In particular, a dimensionless parameter theta is introduced to quantify the migration tendency of predators while they are handling their prey, which gives rise to a family of models connecting two extremes: predators have no inclination to migrate while handling prey (theta = 0) and standard diffusion (theta = 1). Various aspects of the model, including changes in the number and the stability of equilibria and limit cycles, are investigated. We then focus on the key question: "Does spatial structure lead to a substantial damping of the violent oscillations exhibited by many predator-prey models?". It is known that the answer is "yes" if one adopts standard diffusion (theta = 1). However, we present substantial evidence that the answer is "no" if one takes theta = 0. We conclude that the migration submodel is an important constituent of a spatial predator-prey model and that the issue deserves scrutiny, both experimentally and theoretically.

Ecosystem Resilience and Productivity: Are Predictions Possible?

The notion of ecosystem stability has a long history in ecology. The classical question is whether some systems are more stable than others, and if so, what makes them more stable. The stability concept as such has been intensively debated and the confusion about its meaning has led to opposite conclusions about the relationship between food web structure and its dynamic properties (Pimm 1991). A recurrent problem focuses on ecosystem responses to increased productivity. Notwithstanding the far from solved problem of exactly how should be represented in food web models, many attempts have been made to predict how dynamic stability properties change with changes in (Rosenzweig 1971, Nakajima and DeAngelis 1989, DeAngelis 1992, Abrams and Roth 1994, Rosenzweig 1995 for review), how food web structure itself is affected by levels of (Fretwell 1977, Oksanen et al. 1981, DeAngelis et al. 1989, Rosenzweig 1995), and how population interactions within the food web best should be modeled (e.g., the preyvs ratio-dependent debate; Arditi and Ginzburg 1989, Berryman 1992, Ginzburg and Akvakaya 1992, Oksanen et al. 1992, Abrams 1994, Lundberg and Fryxell 1995). Recently, Stone et al. (1996) addressed the relationship between ecosystem resilience and by formally analyzing the return time in single-species models in discrete time and a predator-prey model in continuous time. The return time of the system is the speed at which the equilibrium is approached after disturbance. Ecosystem resilience is therefore inversely proportional to return time. In single-species models, the return time is unambiguously determined by r, the maximum per capita growth rate of the population (given that we follow conventional notation). The maximum per capita growth rate is a quite plausible parameter relating population responses to productivity, although we may wonder (biologically, not mathematically) what happened to population carrying capacity (K). In systems with two or more species, the carrying capacity potentially matters, depending on what model one chooses to work with and how is defined in terms of the parameters of the model. We here extend the work by Stone et al. (1996) and we show that the conclusion about effects on ecosystem resilience is 1) strongly model dependent and 2) dependent on the fundamental assumption of how productivity is defined in terms of the parameters of the model. We will thus reinforce Stone et al.'s suggestion that it will be difficult to find any universal relationship between stability (here resilience) and productivity. Stone et al. (1996) showed that in a two-species predator-prey system, return time (TR) by and large increases as maximum per capita growth rate increases. Given that r is a fair representation of system productivity, this conclusion is not generally valid. To illustrate this, we used two familiar predator-prey models that differ from the model that Stone et al. used (shown in Murray 1990). The first one is the well-known Rosenzweig-MacArthur model

Singular homoclinic bifurcations in tritrophic food chains.

The Rosenzweig-MacArthur food chain model is proved to have homoclinic orbits. The proof is in two steps. First, we use a geometric approach based on singular perturbation and detect singular homoclinic orbits as well as parameter combinations for which these orbits exist. Second, we show, numerically, that for slightly different parameter values there exist also nonsingular homoclinic orbits that tend toward the singular ones when the time responses of the three trophic levels are extremely diversified. The analysis is performed without exploiting too deeply the mathematical structure of the Rosenzweig-MacArthur model. This is done intentionally, to assist readers interested more in the methodology than in the application to food chains.

Yield and Dynamics of Tritrophic Food Chains

Strong relationships between yield and dynamic behavior of tritrophic food chains are pointed out by analyzing the classical Rosenzweig-MacArthur model. On the one hand, food chains are subdivided into undersupplied and oversupplied categories, the first being those in which a marginal increase of nutrient supply to the bottom produces a marginal increase of mean yield at the top. On the other hand, a detailed bifurcation analysis proves that dynamic complexity first increases with nutrient supply (from stationary to a low-frequency cyclic regime and, finally, to chaos) and then decreases (from chaos to a high-frequency cyclic regime). A careful comparison of the two analyses supports the conclusion that food chains cycling at high frequency are oversupplied, while all others are undersupplied. A straightforward consequence of this result is that maximization of food yield requires a chaotic regime. This regime turns out to be very often on the edge of a potential catastrophic collapse of the top component of the food chain. In other words, optimality implies very complex and dangerous dynamics, as intuitively understood long ago for ditrophic food chains by Rosenzweig in his famous article on the paradox of enrichment.

Bioeconomic modelling and fisheries management

This paper examines the stability of the Rosenzweig–MacArthur model distributed to identical discrete habitat patches. Migration between patches is assumed to follow the non-diffusive rule that individuals have a fixed rate of leaving their local habitat patch and migrating to another. Under this non-diffusive migration rule, we found that population dispersal on a non-regular and connected habitat network can both stabilize and destabilize the Rosenzweig–MacArthur model. It is also shown that our non-diffusive migration rule apparently becomes diffusive if the habitat network is regular.

On the Uniqueness of a Limit Cycle for a Predator-Prey System

The uniqueness of a limit cycle for a predator-prey system is proved in this paper. The method used is an improvement of the method used earlier by Cheng.