Effect of hunting cooperation and fear in a predator-prey model

Dynamics of predator-prey systems under the influence of cooperative hunting among predators and the fear thus imposed on the prey population is of great importance from ecological point of view. The role of hunting cooperation and the fear effect in the predator-prey system is gaining considerable attention by the researchers recently. But the study on combined effect of hunting cooperation and fear in the predator-prey system is not yet studied. In the present paper, we investigate the impact of hunting cooperation among predators and predator induced fear in prey population by using the classical predator-prey model. We consider that predator populations cooperate during hunting. We also consider that hunting cooperation induces fear among prey, which has far richer and complex dynamics. We observe that without hunting cooperation, the unique coexistence equilibrium point is globally asymptotically stable. However, an increase in the hunting cooperation induced fear may destabilize the system and produce periodic solution via Hopf-bifurcation. The stability of the Hopf-bifurcating periodic solution is obtained by computing the Lyapunov coefficient. The limit cycles thus obtained may be supercritical or subcritical. We also observe that the system undergoes the Bogdanov-Takens bifurcation in two-parameter space. Further, we observe that the system exhibits backward bifurcation between predator-free equilibrium and coexisting equilibrium. The system also exhibits two different types of bi-stabilities due to subcritical Hopf-bifurcation (between interior equilibrium and stable limit cycle) and backward bifurcation (between predator-free and interior equilibrium points). Further, we observe strong demographic Allee phenomenon in the system. To visualize the dynamical behavior of the system, extensive numerical experiments are performed by using MATLAB and MATCONT softwares.     

Mathematical modeling of the population dynamics of a distinct interactions type system with local dispersal

Distinct biotic interactions in multi-species communities are a ubiquitous force in the natural ecosystem, and this force is an essential determinant of community stability and species coexistence outcomes. We conduct numerical simulations and bifurcation analysis of partial differential equations to gain better understanding and ecological insights into how predation (a), predator handling time (h), and local dispersal affect multi-species community dynamics. This system consists of resource-mutualist-exploiter-competitor interactions and local dispersal. From the inspection of our numerical simulations and co-dimension one bifurcation analysis findings, we discover several critical values that correspond to transcritical bifurcation, subcritical and supercritical Hopf bifurcations. This occurs as we vary the bifurcation parameters a and h in this complex ecological system under symmetric and asymmetric dispersal scenarios. Furthermore, the interplay between these local bifurcation points results in an exciting co-dimension two bifurcations, i.e., Bogdanov-Takens and cusp bifurcation points, respectively, which act as the synchronization points in this complex ecological system. From an ecological viewpoint, we find that (i) the effect of the no-dispersal scenario supports the maintenance of species biodiversity when the predation strength is moderate; (ii) symmetric dispersal induces both subcritical and supercritical Hopf bifurcation and support species diversity for moderate predation strength; and (iii) asymmetric dispersal promotes species diversity as it simplifies the bifurcation changes in dynamics by eliminating the subcritical bifurcations that trigger uncertainty, and this dispersal mechanism mediates species coexistence outcomes. Fundamentally, stable limit cycles have been reported as predator handling time varies in some ecological models; however, we observed in our bifurcation analysis the emergence of the unstable limit cycle as predator handling time changes. We discover that intense predator handling time destabilizes this complex ecological community. In general, our results demonstrate the influential roles of predation, predator handling time, and local dispersal in determining this system’s coexistence dynamics. This knowledge provides a better understanding of species conservation and biological control management.     

Parametric analysis of the ratio-dependent predator–prey model

We present a complete parametric analysis of stability properties and dynamic regimes of an ODE model in which the functional response is a function of the ratio of prey and predator abundances. We show the existence of eight qualitatively different types of system behaviors realized for various parameter values. In particular, there exist areas of coexistence (which may be steady or oscillating), areas in which both populations become extinct, and areas of "conditional coexistence" depending on the initial values. One of the main mathematical features of ratio-dependent models, distinguishing this class from other predator-prey models, is that the Origin is a complicated equilibrium point, whose characteristics crucially determine the main properties of the model. This is the first demonstration of this phenomenon in an ecological model. The model is investigated with methods of the qualitative theory of ODEs and the theory of bifurcations. The biological relevance of the mathematical results is discussed both regarding conservation issues (for which coexistence is desired) and biological control (for which extinction is desired).     

Multiple limit cycles in the standard model of three species competition for three essential resources

We consider the dynamics of the standard model of 3 species competing for 3 essential (non-substitutable) resources in a chemostat using Liebig's law of the minimum functional response. A subset of these systems which possess cyclic symmetry such that its three single-population equilibria are part of a heteroclinic cycle bounding the two-dimensional carrying simplex is examined. We show that a subcritical Hopf bifurcation from the coexistence equilibrium together with a repelling heteroclinic cycle leads to the existence of at least two limit cycles enclosing the coexistence equilibrium on the carrying simplex- the ``inside' one is an unstable separatrix and the ``outside' one is at least semi-stable relative to the carrying simplex. Numerical simulations suggest that there are exactly two limit cycles and that almost every positive solution approaches either the stable limit cycle or the stable coexistence equilibrium, depending on initial conditions. Bifurcation diagrams confirm this picture and show additional features. In an alternative scenario, we show that the subcritical Hopf together with an attracting heteroclinic cycle leads to an unstable periodic orbit separatrix. This research was partially supported by NSF grant DMS 0211614. KY 40292, USA. This author's research was supported in part by NSF grant DMS 0107160     

Impact of Allee effect on an eco-epidemiological system

In this paper, we propose a general ratio-dependent prey-predator model with disease in predator subject to the strong Allee effect in prey. We obtain the complete dynamics of both models: (a) full model with Allee effect; (b) full model without Allee effect. Model (a) may have more than one interior equilibrium point, but model (b) has only one interior equilibrium point. Numerical results reveal that the coexistence of all the populations at the endemic state is possible for both the models. But for the model with Allee effect, the coexistence can be destroyed by an increased supply of alternative food for the predators. It can also be proved that for the full model with Allee effect, the disease can be suppressed under certain parametric conditions. Also by comparing models (a) and (b), we conclude that Allee effect can create or destroy the interior attractor. Finally, we have studied the disease free-submodel (prey and susceptible predator model) with and without Allee effect. The comparative study between these two submodels leads to the following conclusions: 1) In the presence of Allee effect, the number of interior equilibrium points can change from zero to two whereas the submodel without Allee effect has unique interior equilibrium point; 2) Both with and without Allee effect, initial conditions play an important role on the survival and extinction of prey as well as its corresponding predator; 3) In the presence of Allee effect, bi-stability occurs with stable or periodic coexistence of prey and susceptible predator and the extinction of prey and susceptible predator; 4) Allee effect can generate or destroy the interior equilibrium points.     

The discrete Rosenzweig model

Discrete time versions of the Rosenweig predator-prey model are studied by analytic and numerical methods. The interaction of the Hopf bifurcation leading to periodic orbits and the period-doubling bifurcation is investigated. It is shown that for certain choices of the parameters there is stable coexistence of both species together with a local attractor at which the prey is absent.     

Population size dependence, competitive coexistence and habitat destruction

1. Spatial dynamics can lead to coexistence of competing species even with strong asymmetric competition under the assumption that the inferior competitor is a better colonizer given equal rates of extinction. Patterns of habitat fragmentation may alter competitive coexistence under this assumption.
2. Numerical models were developed to test for the previously ignored effect of population size on competitive exclusion and on extinction rates for coexistence of competing species. These models neglect spatial arrangement.
3. Cellular automata were developed to test the effect of population size on competitive coexistence of two species, given that the inferior competitor is a better colonizer. The cellular automata in the present study were stochastic in that they were based upon colonization and extinction probabilities rather than deterministic rules.
4. The effect of population size on competitive exclusion at the local scale was found to have little consequence for the coexistence of competitors at the metapopulation (or landscape) scale. In contrast, population size effects on extinction at the local scale led to much reduced landscape scale coexistence compared to simulations not including localized population size effects on extinction, especially in the cellular automata models. Spatially explicit dynamics of the cellular automata vs. deterministic rates of the numerical model resulted in decreased survival of both species. One important finding is that superior competitors that are widespread can become extinct before less common inferior competitors because of limited colonization.
5. These results suggest that population size–extinction relationships may play a large role in competitive coexistence. These results and differences are used in a model structure to help reconcile previous spatially explicit studies which provided apparently different results concerning coexistence of competing species.     

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 positive coexistence of a nonlinear elliptic biological interacting model

The purpose of this note is to give a necessary and sufficient condition for the coexistence of positive solutions to a rather general type of elliptic predator-prey system of the Dirichlet problem on the bounded domain omega when omega is a subset of Rn is large. The result is that the partial differential equation system possesses positive coexistence if and only if the corresponding ordinary differential equation system has positive equilibrium, the positive constant states. This result thus yields an algebraically computable criterion for the positive coexistence of predator and prey in many biological models.     

Critical slowing down as an indicator of transitions in two-species models

Transitions in ecological systems often occur without apparent warning, and may represent shifts between alternative persistent states. Decreasing ecological resilience (the size of the basin of attraction around a stable state) can signal an impending transition, but this effect is difficult to measure in practice. Recent research has suggested that a decreasing rate of recovery from small perturbations (critical slowing down) is a good indicator of ecological resilience. Here we use analytical techniques to draw general conclusions about the conditions under which critical slowing down provides an early indicator of transitions in two-species predator-prey and competition models. The models exhibit three types of transition: the predator-prey model has a Hopf bifurcation and a transcritical bifurcation, and the competition model has two saddle-node bifurcations (in which case the system exhibits hysteresis) or two transcritical bifurcations, depending on the parameterisation. We find that critical slowing down is an earlier indicator of the Hopf bifurcation in predator-prey models in which prey are regulated by predation rather than by intrinsic density-dependent effects and an earlier indicator of transitions in competition models in which the dynamics of the rare species operate on slower timescales than the dynamics of the common species. These results lead directly to predictions for more complex multi-species systems, which can be tested using simulation models or real ecosystems.     

一类食饵种群中具有传染病的捕食-被捕食系统

对疾病仅在食饵种群传播的有比例依赖的捕食-被捕食系统的动力学进行了分析,给出了每个平衡点附近系统的性态,定义了决定疾病灭绝和成为地方病的阁值R_0．得出的结论是:在比例依赖的捕食-被捕食系统中,染病食饵种群可以充当一个生物控制量,以抑制种群的绝灭．     

Transient dynamics limit the effectiveness of keystone predation in bringing about coexistence

We analyze the transient dynamics of simple models of keystone predation, in which a predator preferentially consumes the dominant of two (or more) competing prey species. We show that coexistence is unlikely in many systems characterized both by successful invasion of either prey species into the food web that lacks it and by a stable equilibrium with high densities of all species. Invasion of the predator-resistant consumer species often causes the resident, more vulnerable prey to crash to such low densities that extinction would occur for many realistic population sizes. Subsequent transient cycles may entail very low densities of the predator or of the initially successful invader, which may also preclude coexistence of finite populations. Factors causing particularly low minimum densities during the transient cycles include biotic limiting resources for the prey, limited resource partitioning between the prey, a highly efficient predator with relatively slow dynamics, and a vulnerable prey whose population dynamics are rapid relative to the less vulnerable prey. Under these conditions, coexistence of competing prey via keystone predation often requires that the prey's competitive or antipredator characteristics fall within very narrow ranges. Similar transient crashes are likely to occur in other food webs and food web models.     

On the importance of dimensionality of space in models of space-mediated population persistence

Spatially explicit models have become widely used in today's mathematical ecology to study persistence of populations. For the sake of simplicity, population dynamics is often analyzed with 1-D models. An important question is: how adequate is such 1-D simplification of 2-D (or 3-D) dynamics for predicting species persistence. Here we show that dimensionality of the environment can play a critical role in the persistence of predator-prey interactions. We consider 1-D and 2-D dynamics of a predator-prey model with the prey growth damped by the Allee effect. We show that adding a second space coordinate into the 1-D model results in a pronounced increase of size of the domain in the parametric space where predator-prey coexistence becomes possible. This result is due to the possibility of formation of a number of 2-D patterns, which is impossible in the 1-D model. The 1-D and the 2-D models exhibit different qualitative responses to variations of system parameters. We show that in ecosystems having a narrow width (e.g. mountain valleys, vegetation patterns along canals in dry areas, etc.), extinction of species is more probable compared to ecosystems having a pronounced second dimension. In particular, the width of a long narrow natural reserve should be large enough to guarantee nonextinction of species via interaction of 2-D population patches.     

A ratio-dependent food chain model and its applications to biological control

While biological controls have been successfully and frequently implemented by nature and human, plausible mathematical models are yet to be found to explain the often observed deterministic extinctions of both pest and control agent in such processes. In this paper we study a three trophic level food chain model with ratio-dependent Michaelis-Menten type functional responses. We shall show that this model is rich in boundary dynamics and is capable of generating such extinction dynamics. Two trophic level Michaelis-Menten type ratio-dependent predator-prey system was globally and systematically analyzed in details recently. A distinct and realistic feature of ratio-dependence is its capability of producing the extinction of prey species, and hence the collapse of the system. Another distinctive feature of this model is that its dynamical outcomes may depend on initial populations levels. Theses features, if preserved in a three trophic food chain model, make it appealing for modelling certain biological control processes (where prey is a plant species, middle predator as a pest, and top predator as a biological control agent) where the simultaneous extinctions of pest and control agent is the hallmark of their successes and are usually dependent on the amount of control agent. Our results indicate that this extinction dynamics and sensitivity to initial population levels are not only preserved, but also enriched in the three trophic level food chain model. Specifically, we provide partial answers to questions such as: under what scenarios a potential biological control may be successful, and when it may fail. We also study the questions such as what conditions ensure the coexistence of all the three species in the forms of a stable steady state and limit cycle, respectively. A multiple attractor scenario is found.     

Competition and stoichiometry: coexistence of two predators on one prey

The competitive exclusion principle (CEP) states that no equilibrium is possible if n species exploit fewer than n resources. This principle does not appear to hold in nature, where high biodiversity is commonly observed, even in seemingly homogenous habitats. Although various mechanisms, such as spatial heterogeneity or chaotic fluctuations, have been proposed to explain this coexistence, none of them invalidates this principle. Here we evaluate whether principles of ecological stoichiometry can contribute to the stable maintenance of biodiverse communities. Stoichiometric analysis recognizes that each organism is a mixture of multiple chemical elements such as carbon (C), nitrogen (N), and phosphorus (P) that are present in various proportions in organisms. We incorporate these principles into a standard predator-prey model to analyze competition between two predators on one autotrophic prey. The model tracks two essential elements, C and P, in each species. We show that a stable equilibrium is possible with two predators on this single prey. At this equilibrium both predators can be limited by the P content of the prey. The analysis suggests that chemical heterogeneity within and among species provides new mechanisms that can support species coexistence and that may be important in maintaining biodiversity.     

Bifurcation Analysis of Models with Uncertain Function Specification: How Should We Proceed?

When we investigate the bifurcation structure of models of natural phenomena, we usually assume that all model functions are mathematically specified and that the only existing uncertainty is with respect to the parameters of these functions. In this case, we can split the parameter space into domains corresponding to qualitatively similar dynamics, separated by bifurcation hypersurfaces. On the other hand, in the biological sciences, the exact shape of the model functions is often unknown, and only some qualitative properties of the functions can be specified: mathematically, we can consider that the unknown functions belong to a specific class of functions. However, the use of two different functions belonging to the same class can result in qualitatively different dynamical behaviour in the model and different types of bifurcation. In the literature, the conventional way to avoid such ambiguity is to narrow the class of unknown functions, which allows us to keep patterns of dynamical behaviour consistent for varying functions. The main shortcoming of this approach is that the restrictions on the model functions are often given by cumbersome expressions and are strictly model-dependent: biologically, they are meaningless. In this paper, we suggest a new framework (based on the ODE paradigm) which allows us to investigate deterministic biological models in which the mathematical formulation of some functions is unspecified except for some generic qualitative properties. We demonstrate that in such models, the conventional idea of revealing a concrete bifurcation structure becomes irrelevant: we can only describe bifurcations with a certain probability. We then propose a method to define the probability of a bifurcation taking place when there is uncertainty in the parameterisation in our model. As an illustrative example, we consider a generic predator–prey model where the use of different parameterisations of the logistic-type prey growth function can result in different dynamics in terms of the type of the Hopf bifurcation through which the coexistence equilibrium loses stability. Using this system, we demonstrate a framework for evaluating the probability of having a supercritical or subcritical Hopf bifurcation.     

Bifurcating periodic solutions for a class of age-structured predator-prey systems

A system of mixed integrodifferential and partial differential equations for an agestructured predator-prey system is studied here. The predator eats all ages of prey, but more of the very young and very old than of the intermediate ages. The existence of periodic solutions corresponding to stable coexistence is proved for a suitable range of parameters by bifurcation theory.     

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.     

The role of variability and risk on the persistence of shared-enemy,predator-prey assemblages

The role of indirect effects such as apparent competition in structuring predator-prey assemblages has recently received empirical attention. That one prey species can be excluded by the impact of a shared-enemy contrasts with the known diversity of multispecies predator-prey interactions. Here, the role of predator foraging among patches of two different prey species is examined as a mechanism that can mediate coexistence in multispecies prey-predator assemblages. Specifically, models of host-parasitoid interactions are constructed to analyse how different types of aggregative behaviour (generated by host-dependent and host-independent responses) affect persistence of the assemblage. How the distribution of hosts and the response of the parasitoid to these distributions can influence coexistence is shown. A generic explanation for coexistence suggests that it is the variability rather than the precise functional relationship that is critical for coexistence under shared-enemy interactions.