Stability regions and transition phenomena for harvested predator-prey systems

Summary We analyze the global behaviour of a predator-prey system under constant-rate predator harvesting, showing how to classify the possibilities and determine the region of asymptotic stability by a combination of relatively elementary theoretical methods and computer simulations.Research sponsored in part by the National Research Council of Canada. Grant No. A-3138     

Stability regions in predator-prey systems with constant-rate prey harvesting

Summary We analyze the global behavior of a predator-prey system, modelled by a pair of non-linear ordinary differential equations, under constant-rate prey harvesting. By methods analogous to those used to study predator harvesting, we characterize the theoretically possible structures and transitions. With the aid of a computer simulation we construct examples to show which of these transitions can be realized in a biologically plausible model.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Research Council of Canada, Grant No. 67-3138.     

Persistence of predator-prey systems in an uncertain environment

Summary The time derivatives of prey and predator populations are assumed to satisfy a set of inequalities, instead of a precise differential equation, reflecting an uncertain environmental and/or lack of knowledge by the modeler. A system of differential equations is found whose solution gives the boundary of a persistent set, which is positive flow invariant for any system satisfying the inequalities. Conditions are given for the persistent set to be bounded away from both axes, which show that resonance effects cannot drive either predator or prey to extinction if that does not happen for an autonomous system satisfying the inequalities. In general predator-prey systems are more persistent when there is strong asymptotic stability, when there is correlation between prey and predator dynamics, when the effect of perturbations is density dependent, and are more persistent under perturbations of the prey than of the predator.     

Encounters in predator-prey systems: a simple discrete model

Predator-prey systems are often described by exploitation models. These models can mimic experimental data very accurately, but it is sometimes difficult to realize the relationships between the models and the behavior of individual predator and prey animals. A simple discrete model is proposed here that tries to elucidate the connections between: the animals' movements, the predator/prey encounters; and the dynamics in the system as globally represented by the exploitation models. In these models, the term "area of discovery" plays an essential role. This term is shown to be a predictable coefficient that is composed of measurable physical properties of the analyzed predator-prey system. The model takes into account that predators and prey in experimental systems often do not search randomly but prefer some parts of the test area. The model is applied to the mite system Phytoseiulus persimilis/Tetranychus urticae under simple artificial conditions.     

Constant-rate stocking of predator-prey systems

We examine the qualitative effects of constant-rate stocking of either or both species in a predator-prey system. The hypotheses are made as mild as possible so that several types of systems with different qualitative alternatives may be studied.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Research Council of Canada under Grant No. A-3138.The authors wish to thank Mr. Al Mackenzie for drawing the numerous figures which appear in this paper.     

A predator-prey reaction-diffusion system with nonlocal effects

We consider a predator-prey system in the form of a coupled system of reaction-diffusion equations with an integral term representing a weighted average of the values of the prey density function, both in past time and space. In a limiting case the system reduces to the Lotka Volterra diffusion system with logistic growth of the prey. We investigate the linear stability of the coexistence steady state and bifurcations occurring from it, and expressions for some of the bifurcating solutions are constructed. None of these bifurcations can occur in the degenerate case when the nonlocal term is in fact local.     

Coexistence properties of some predator-prey systems under constant rate harvesting and stocking

The global behaviour of a class of predator-prey systems, modelled by a pair of non-linear ordinary differential equations, under constant rate harvesting and/or stocking of both species, is presented. Theoretically possible structures and transitions are developed and validated by computer simulations. The results are presented as transition loci in the F-G (prey harvest rate-predator harvest rate) plane.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 and in part by NSERC of Canada, Grant No. 67-3138The authors wish to thank Mr. Al MacKenzie of the Department of Electrical Engineering, University of British Columbia, for preparing the figures in this paper.     

Synchronisation and stability in river metapopulation networks

Spatial structure in landscapes impacts population stability. Two linked components of stability have large consequences for persistence: first, statistical stability as the lack of temporal fluctuations; second, synchronisation as an aspect of dynamic stability, which erodes metapopulation rescue effects. Here, we determine the influence of river network structure on the stability of riverine metapopulations. We introduce an approach that converts river networks to metapopulation networks, and analytically show how fluctuation magnitude is influenced by interaction structure. We show that river metapopulation complexity (in terms of branching prevalence) has nonlinear dampening effects on population fluctuations, and can also buffer against synchronisation. We conclude by showing that river transects generally increase synchronisation, while the spatial scale of interaction has nonlinear effects on synchronised dynamics. Our results indicate that this dual stability – conferred by fluctuation and synchronisation dampening – emerges from interaction structure in rivers, and this may strongly influence the persistence of river metapopulations.     

Models for the effect of toxicant in single-species and predator-prey systems

Models of single-species and predator-prey systems in a polluted closed environment are developed and partially analyzed. Three cases are considered: a single influx of toxicant, a constant influx of toxicant, and a periodic pollution of the environment. In the case of single-species growth we are able to determine some local and global dynamics. In the case of predator-prey systems, we investigate the existence of steady states for a small constant influx of toxicant.On leave from Department of Mathematics, Indian Institute of Technology, Kanpur, IndiaResearch partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. NSERC A4823     

Global stability of a predator-prey system

In this paper we derive a result to ensure the global stability of a predator-prey system. The method used is quite general and may have applications to other situations.Works were partially supported by the National Science Council of the Republic of China     

Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations

We investigate the emergence of spatio-temporal patterns in ecological systems. In particular, we study a generalized predator-prey system on a spatial domain. On this domain diffusion is considered as the principal process of motion. We derive the conditions for Hopf and Turing instabilities without specifying the predator-prey functional responses and discuss their biological implications. Furthermore, we identify the codimension-2 Turing-Hopf bifurcation and the codimension-3 Turing-Takens-Bogdanov bifurcation. These bifurcations give rise to complex pattern formation processes in their neighborhood. Our theoretical findings are illustrated with a specific model. In simulations a large variety of different types of long-term behavior, including homogenous distributions, stationary spatial patterns and complex spatio-temporal patterns, are observed.     

Multiple limit cycles in predator-prey models

A series of one-predator one-prey models are studied using two parameter Hopf bifurcation techniques which allow the determination of two periodic orbits. The biological implications of the results, in terms of domains of attraction and multiple stable states, are discussed.     

Stabilizing dispersal delays in predator-prey metapopulation models

Time delays produced by dispersal are shown to stabilize Lotka-Volterra predator-prey models. The models are formulated as integrodifferential equations that describe local predator-prey dynamics and either intrapatch or interpatch dispersal. Dispersing individuals may (or may not) differ in the duration of their trips; these differences are captured via a distributed delay in the models. Our results include those of previous studies as special cases and show that the stabilizing effect continues to operate when the dispersal process is modeled more realistically.     

Habitat shape, metapopulation processes and the dynamics of multispecies predator-prey interactions

1. The effects of habitat shape, connectivity and the metapopulation processes of persistence and extinction are explored in a multispecies resource-consumer interaction. 2. The spatial dynamics of the indirect interaction between two prey species (Callosobruchus chinensis, Callosobruchus maculatus) and a predator (Anisopteromalus calandrae) are investigated and we show how the persistence time of this interaction is altered in different habitat configurations by the presence of an apparent competitor. 3. Habitat structure has differential effects on the dynamics of the resource-consumer interaction. Across all habitat types, the pairwise interaction between C. chinensis and A. calandrae is highly prone to extinction, while the interaction between C. maculatus and A. calandrae shows sustained long-term fluctuations. Contrary to expectations from theory, habitat shape has no significant effect on persistence time of the full, three-species resource-consumer assemblage. 4. A stochastic metapopulation model for a range of habitat configurations, incorporating different forms of regulatory processes, highlights that it is the spatially explicit population dynamics rather than the shape of the metapopulation that is the principal determinant of interaction persistence time.     

No-escape regions and oscillations in second order predator-prey recurrences

We are concerned with the second order recurrence x _n+1 = x _n f(x _n, y _n), y _n+1 = y _n g(x _n, y _n), where n N ₀, x ₀ > 0, y ₀ > 0, and the reproduction rates f and g simulate predator-prey interaction. Under conditions on the sign of f – 1 and g–1 we show the existence of a nontrivial no-escape region D, i.e. a compact set D {(x, y): x > 0, y > 0} which is invariant under the recurrence and has the property that every sequence enters D after finitely many steps. Under further conditions on the shape of the isoclines {f = 1} and {g = 1} and on the stationary points {f = 1} {g = 1} we are able to show the existence of sustained oscillations.This work has been supported by the Deutsche Forschungsgemeinschaft     

Some results on global stability of a predator-prey system

In this paper we derive some results to ensure the global stability of a predator-prey system. The results cover most of the models which have been proposed in the ecological literature for predator-prey systems. The first result is very geometric and it is very easy to check from the graph of prey and predator isoclines. The second one is purely algebraic, however, it covers the defects of the first one especially in dealing with Holling's type-3 functional response in some sense. We also discuss the global stability of Kolmogorov's model. Some examples are presented in the discussion section.Works partially supported by the National Science Council of the Republic of China     

An acarine predator-prey metapopulation system inhabiting greenhouse cucumbers

The two-spotted spider mite ( Tetranychus urticae ) is a serious pest on greenhouse cucumbers, but can be controlled by the phytoseiid predator Phytoseiulus persimilis. The two mite species exhibit considerable fluctuations in overall population densities but within acceptable limits. The system appears to be persistent at a regional (greenhouse) scale in spite of frequent local extinctions (e.g. at individual plants). Experimental evidence indicates that the mites form a metapopulation system characterized by 'shifting mosaic' dynamics. A stochastic simulation model is used to analyse the role played by dispersal in the dynamics and persistence of the system. It shows that demographic stochasticity generates sufficient endogenous 'noise' to counteract the synchronizing effect of density-dependent dispersal, provided dispersal rates are not too high and the system is not too small. Low dispersal rates, on the other hand, increase the risk of local outbreaks of spider mites that may cause destruction of plants.