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.     

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.     

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     

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     

Effects of behavioral adaptation on a predator-prey model

The Volterra-Lotka predator-prey equations are modified so that the predator's ability to utilize the prey varies in proportion to the average number of encounters between the two species in the past. The behavior of this adaptive system is then described in terms of three parameters — the carrying capacity of the prey, the relative death rate of the predator, and the predator's memoryspan. The most stable situation is shown to occur when the carrying capacity of the prey is large, the predator's death rate is close to zero, and the predator is able to adapt quickly to changing levels of prey density.     

Studying metapopulation effects in predator-prey systems

It has been suggested that many predator-prey systems persist, despite unstable local interactions, due to metapopulation processes: movement of individuals among largely independent local populations. I review 13 possible examples of this phenomenon all I could find in the literature—and find that each either lacks convincing data or is not a true metapopulation. Most of the examples rely on evidence of local extinction and recolonization, only a few using direct experimental methods; I discuss the advantages and disadvantages of these methods, as well as alternatives.
I also consider the ways these systems deviate from having pure metapopulation structures, and conclude that most large-scale spatial population structures will not fit cleanly into a metapopulation vs. within-population dichotomy, but rather combine features of both. This will necessitate use of powerful and focused methodology (in particular, experimentation) to directly describe movement rates and patterns, rather than use of crude observational data (e.g. extinctions) to make inferences about movement.     

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.     

A detailed study of the Beddington-DeAngelis predator-prey model

The present investigation accounts for the influence of intra-specific competition among predators in the original Beddington-DeAngelis predator-prey model. We offer a detailed mathematical analysis of the model to describe some of the significant results that may be expected to arise from the interplay of deterministic and stochastic biological phenomena and processes. In particular, stability (local and global) and bifurcation (Saddle-node, Transcritical, Hopf-Andronov, Bogdanov-Takens) analysis of this model are conducted. Corresponding results from previous well known predator-prey models are compared with the current findings. Nevertheless, we also allow this model in stochastic environment with the influences of both, uncorrelated “white” noise and correlated “coloured” noise. This showing that competition among the predator population is beneficial for a number of predator-prey models by keeping them stable around its positive interior equilibrium (i.e. when both populations co-exist), under environmental stochasticity. Comparisons of these findings with the results of some earlier related investigations allow the general conclusion that predator intra-species competition benefits the predator-prey system under both deterministic and stochastic environments. Finally, an extended discussion of the ecological implications of the analytical and numerical results concludes the paper.     

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     

捕食者-猎物关系的理论和应用研究

捕食者和猎物相互关系的研究,长期以来一直是动物生态学研究的中心课题之一。这种研究可区分为3种类型:第一是理论研究,即组建各种数学模型以便模拟捕食者和猎物间的相互关系;第二是实验种群研究,通常是在实验室内选用原生动物和节肢动物实验种群对捕食一猎物间的动态关系进行观察,并将观察结果与理论模型进行比较;第三是在田间对自然种群中的捕食者-猎物关系进行研究,并利用从理论研究和实验种群研究中所总结出来的各种基本原理对观察资料进行分析。在应用生态学领域中,常常靠引进新的更加有效的天敌来控制各种害虫。这些实际工作有些已获得成功,也有不少未取得预期效果,不管成功与否,它们都可被看作是对捕食者-猎物关系所进行的田间实验.     

Stability,sensitivity and uncertainty rates in the flow equations of ecological models

In previous work by the authors about dynamic system modeling, basic ecosystems concepts and their application to ecological modeling theory were formalized. Measuring how a variable effects certain processes leads to improvements in dynamic systems modelling and facilitates the author’s study of system diversity in which model sensitivity is a key theme. Initially, some variables and their numeric data are used for modeling. Predictions from the constructed models depend on these data. Generic study of sensitivity aims to show to what degree model behavior is altered by modification of some specific data. If small variations cause important modifications in the model’s global behavior then the model is very sensitive in relation to the variables used. If uncertain systems are considered, then it is important to submit the system to extreme situations and analyze its behavior. In this article, some indexes of uncertainty will be defined in order to determine the variables' influence in the case of extreme changes. This will permit analysis of the system’s sensitivity in relation to several simulations.     

On constant effort harvesting and stocking in a class of predator-prey systems

We analyze the global behaviour of predator-prey systems under constant effort harvesting and stocking of either or both species. Mathematically this is equivalent to the behaviour of an unharvested system with different parameters; in particular every solution tends either to an equilibrium or to a limit cycle, and there is no possibility of finite-time extinction of either species if both populations are positive initially. The dependence of the behaviour on the harvesting efforts is discussed.     

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.     

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     

Effects of prey refuges on a predator-prey model with a class of functional responses: The role of refuges

In this paper, the effects of refuges used by prey on a predator-prey interaction with a class of functional responses are studied by using the analytical approach. The refuges are considered as two types: a constant proportion of prey and a fixed number of prey using refuges. We will evaluate the effects with regard to the local stability of the interior equilibrium point, the values of the equilibrium density and the long-term dynamics of the interacting populations. The results show that the effects of refuges used by prey increase the equilibrium density of prey population while decrease that of predators. It is also proved that the effects of refuges can stabilize the interior equilibrium point of the considered model, and destabilize it under a very restricted set of conditions which is disagreement with previous results in this field.     

Effects of a disease affecting a predator on the dynamics of a predator-prey system

We study the effects of a disease affecting a predator on the dynamics of a predator-prey system. We couple an SIRS model applied to the predator population, to a Lotka-Volterra model. The SIRS model describes the spread of the disease in a predator population subdivided into susceptible, infected and removed individuals. The Lotka-Volterra model describes the predator-prey interactions. We consider two time scales, a fast one for the disease and a comparatively slow one for predator-prey interactions and for predator mortality. We use the classical “aggregation method” in order to obtain a reduced equivalent model. We show that there are two possible asymptotic behaviors: either the predator population dies out and the prey tends to its carrying capacity, or the predator and prey coexist. In this latter case, the predator population tends either to a “disease-free” or to a “disease-endemic” state. Moreover, the total predator density in the disease-endemic state is greater than the predator density in the “disease-free” equilibrium (DFE).     

Regime changes in ecological systems: an information theory approach

We present our efforts at developing an ecological system index using information theory. Specifically, we derive an expression for Fisher Information based on sampling of the system trajectory as it evolves in the space defined by the state variables of the system, i.e. its state space. The Fisher Information index, as we have derived it, is a measure of system order, and captures the characteristic variation in speed and acceleration along the system's periodic steady-state trajectories. When calculated repeatedly over the system period, this index tracks steady states and transient behavior. We believe that such an index could be useful in detecting system 'flips' associated with a regime change, i.e. determining when systems are in a transient between one steady state and another. We illustrate the concepts using model ecosystems.