Dynamics of predator and modular prey: effects of module consumption on stability of prey–predator system

In traditional models of predator–prey population dynamics, it is usually assumed that consumed prey are immediately removed from the population. However, in plant–herbivore interactions, damaged plants are generally alive after attacks by herbivores. This can result in successive or simultaneous attacks by multiple predators on a single prey item (here, the term prey is expanded to include plants). We constructed a mathematical model with two time scales, taking into account predation processes within a generation, considering post‐predation survival and the modularity of prey. We assumed that a prey item can be divided into modules and that it can be fed on by multiple predators or parasitized by multiple parasites at the same time. The model includes two essential factors: the modularity of prey for predators (n) and the detaching/attaching ratio of predators to prey (ε). Based on the formulae, we revealed a general property of realistic dynamics in plant–herbivore and host–parasite interactions. The analysis showed that the model could be approximated by models with the type I, type II or Beddington–DeAngelis functional responses by taking appropriate limits to the situations. When modularity is low or the detaching/attaching ratio is high, population dynamics tend to be stabilized. These stabilizing effects may be related to interference competition among predator individuals or increases in free prey modules and free predator individuals. In the limit of high modularity, the ratio of the attached prey modules to the total prey modules becomes negligible and the dynamics tend to be destabilized. However, if quantity and quality of prey modules are negatively correlated, the equilibrium is likely to be stabilized at high modularity as long as it remains feasible. These results suggest that considering post‐predation survival and modularity of prey is crucial to understand predator–prey interactions.     

Reversed predator–prey cycles are driven by the amplitude of prey oscillations

Ecoevolutionary feedbacks in predator–prey systems have been shown to qualitatively alter predator–prey dynamics. As a striking example, defense–offense coevolution can reverse predator–prey cycles, so predator peaks precede prey peaks rather than vice versa. However, this has only rarely been shown in either model studies or empirical systems. Here, we investigate whether this rarity is a fundamental feature of reversed cycles by exploring under which conditions they should be found. For this, we first identify potential conditions and parameter ranges most likely to result in reversed cycles by developing a new measure, the effective prey biomass, which combines prey biomass with prey and predator traits, and represents the prey biomass as perceived by the predator. We show that predator dynamics always follow the dynamics of the effective prey biomass with a classic ¼‐phase lag. From this key insight, it follows that in reversed cycles (i.e., ¾‐lag), the dynamics of the actual and the effective prey biomass must be in antiphase with each other, that is, the effective prey biomass must be highest when actual prey biomass is lowest, and vice versa. Based on this, we predict that reversed cycles should be found mainly when oscillations in actual prey biomass are small and thus have limited impact on the dynamics of the effective prey biomass, which are mainly driven by trait changes. We then confirm this prediction using numerical simulations of a coevolutionary predator–prey system, varying the amplitude of the oscillations in prey biomass: Reversed cycles are consistently associated with regions of parameter space leading to small‐amplitude prey oscillations, offering a specific and highly testable prediction for conditions under which reversed cycles should occur in natural systems.     

Irreversible prey diapause as an optimal strategy of a physiologically extended Lotka–Volterra model

We propose an optimal control framework to describe intra-seasonal predator–prey interactions, which are characterized by a continuous-time dynamical model comprising predator and prey density, as well as the energy budget of the prey over the length of a season. The model includes a time-dependent decision variable for the prey, representing the portion of the prey population in time that is active, as opposed to diapausing (a state of physiological rest). The predator follows autonomous dynamics and accordingly it remains active during the season. The proposed model is a generalization of the classical Lotka–Volterra predator–prey model towards non-autonomous dynamics that furthermore includes the effect of an energy variable. The model has been inspired by a specific biological system of predatory mites (Acari: Phytoseiidae) and prey mites (so-called fruit-tree red spider mites) (Acari: Tetranychidae) that feed on leaves of apple trees—its parameters have been instantiated based on laboratory and field studies. The goal of the work is to understand the decisions of the prey mites to enter diapause (a state of physiological rest) given the dynamics of the predatory mites: this is achieved by solving an optimization problem hinging on the maximization of the prey population contribution to the next season. The main features of the optimal strategy for the prey are shown to be that (1) once in diapause, the prey does not become active again within the same season and hence diapause is an irreversible process; (2) for the vast majority of parameter space, the portion of prey individuals entering diapause within the season does not decrease in time; (3) with an increased number of predators, the optimal population strategy for the prey is to start diapause earlier and to enter diapause more gradually. This optimal population strategy will be studied for its ESS properties in a sequel to the work presented in this article.     

Prey-Dependent Mortality Rate: A Critical Parameter in Microbial Models

Protozoa are key components of a wide range of ecosystems, but ecological models that incorporate these microbes often suffer from poor parameterisation, specifically of top-level predator loss rates. We (1) suggest that top-level predator mortality is prey-dependent, (2) provide a novel approach to assess this response, and (3) illustrate the ecological relevance of these findings. Ciliates, Paramecium caudatum (prey) and Didinium nasutum (predator), were used to evaluate predator mortality at varying prey levels. To assess mortality, multiple (>100) predators were individually examined (in 2-ml wells), daily (for 3 days), between 0 and 120 preys ml⁻¹. Data were used to determine non-linear mortality and growth responses over a range of prey abundances. The responses, plus literature data were then used to parameterise a predator–prey model, based on the Rosenzweig–MacArthur structure. The model assessed the impact of variable and three levels of constant (high, average and low) mortality rates on P. caudatum–D. nasutum population dynamics. Our method to determine variable mortality rate revealed a strong concave decline in mortality with increasing prey abundance. The model indicated: (1) high- and low-constant mortality rates yielded dynamics that deviate substantially from those obtained from a variable rate; (2) average mortality rate superficially produced dynamics similar to the variable rate, but there were differences in the period of predator–prey cycles, and the lowest abundance of prey and predators (by ~2 orders of magnitude). The differences between incorporating variable and constant mortality rate indicate that including a variable rate could substantially improve microbial-based ecological models.     

What Happens When Predators Do Not Completely Consume Their Prey?

A mathematical model is presented for the dynamics of predator-prey interactions when predators do not consume prey (or clumps of prey) in their entirety. Using a combination of analytical and numerical methods, I demonstrate that predator-mediated changes in the distribution of intact and partially consumed prey can affect the outcome of competition between predators in unexpected ways. In some cases, two predators can coexist on a single prey species owing to tradeoffs between the ability to consume prey completely and other competitive abilities. In other cases, predators exhibit frequency-dependent dynamics in which the first predator to occupy the habitat can prevent the other from invading. Conditions for stable coexistence usually expand if the larger predator scatters uneaten prey parts, if prey renewal includes both small and large items, or if the predator with the smaller retrieval capacity is poor at catching intact prey relative to the other predator.     

An eco-epidemiological system with infected prey and predator subject to the weak Allee effect

In this article, we propose a general prey–predator model with disease in prey and predator subject to the weak Allee effects. We make the following assumptions: (i) infected prey competes for resources but does not contribute to reproduction; and (ii) in comparison to the consumption of the susceptible prey, consumption of infected prey would contribute less or negatively to the growth of predator. Based on these assumptions, we provide basic dynamic properties for the full model and corresponding submodels with and without the Allee effects. By comparing the disease free submodels (susceptible prey–predator model) with and without the Allee effects, we conclude that the Allee effects can create or destroy the interior attractors. This enables us to obtain the complete dynamics of the full model and conclude that the model has only one attractor (only susceptible prey survives or susceptible-infected coexist), or two attractors (bi-stability with only susceptible prey and susceptible prey–predator coexist or susceptible prey-infected prey coexists and susceptible prey–predator coexist). This model does not support the coexistence of susceptible-infected-predator, which is caused by the assumption that infected population contributes less or are harmful to the growth of predator in comparison to the consumption of susceptible prey.     

Dynamics and responses to mortality rates of competing predators undergoing predator-prey cycles

Two or more competing predators can coexist using a single homogeneous prey species if the system containing all three undergoes internally generated fluctuations in density. However, the dynamics of species that coexist via this mechanism have not been extensively explored. Here, we examine both the nature of the dynamics and the responses of the mean densities of each predator to mortality imposed upon it or its competitor. The analysis of dynamics uncovers several previously undescribed behaviors for this model, including chaotic fluctuations, and long-term transients that differ significantly from the ultimate patterns of fluctuations. The limiting dynamics of the system can be loosely classified as synchronous cycles, asynchronous cycles, and chaotic dynamics. Synchronous cycles are simple limit cycles with highly positively correlated densities of the two predator species. Asynchronous cycles are limit cycles, frequently of complex form, including a significant period during which prey density is nearly constant while one predator gradually, monotonically replaces the other. Chaotic dynamics are aperiodic and generally have intermediate correlations between predator densities. Continuous changes in density-independent mortality rates often lead to abrupt transitions in mean population sizes, and increases in the mortality rate of one predator may decrease the population size of the competing predator. Similarly, increases in the immigration rate of one predator may decrease its own density and increase the density of the other predator. Proportional changes in one predator's birth and death rate functions can have significant effects on the dynamics and mean densities of both predator species. All of these responses to environmental change differ from those observed when competitors coexist stably as the result of resource (prey) partitioning. The patterns described here occur in many other competition models in which there are cycles and differences in the linearity of the responses of consumers to their resources.     

Influence of consumer mutual interference on the stabilization of a three-level trophic system

In this paper, we present a three-level (food–prey–predator) trophic food chain which includes consumer mutual interference (MIF). In contrast with other analyses, we consider the effect of both prey and predator MIF on the dynamics of a three-level trophic system. MIF is generally considered to exert a stabilizing effect on population dynamics based on the predator–prey model. However, results from analytical and numerical simulations utilizing a simple three-species food chain model suggest that while the addition of prey MIF to the model provides a stabilizing influence, as the chaotic dynamics collapse to a stable steady state, adding only predator MIF to the model can only stabilize the system at intermediate MIF values. The three-species trophic food chain is also stabilized when combination of both prey and predator MIF is added to the model. Our work serves to provide insight into the effects of MIF in the real world.     

Variation in foraging success among predators and its implications for population dynamics

The effects of the expected predation rate on population dynamics have been studied intensively, but little is known about the effects of predation rate variability (i.e., predator individuals having variable foraging success) on population dynamics. In this study, variation in foraging success among predators was quantified by observing the predation of the wolf spider Pardosa pseudoannulata on the cricket Gryllus bimaculatus in the laboratory. A population model was then developed, and the effect of foraging variability on predator–prey dynamics was examined by incorporating levels of variation comparable to those quantified in the experiment. The variability in the foraging success among spiders was greater than would be expected by chance (i.e., the random allocation of prey to predators). The foraging variation was density‐dependent; it became higher as the predator density increased. A population model that incorporates foraging variation shows that the variation influences population dynamics by affecting the numerical response of predators. In particular, the variation induces negative density‐dependent effects among predators and stabilizes predator–prey dynamics.     

Individual variation in prey choice in a predator-prey community

One predator-two prey community models are studied with an emphasis on individual variation in predator behavior. The predator behaves according to a well-known prey choice model. The behavioral model predicts that predators should always attack the primary prey (more profitable prey of the two), but only attack the alternative prey (less profitable prey of the two) when the density of the primary prey is below a threshold density. The predator that accepts the alternative prey does not discriminate between the primary and alternative prey (all-or-nothing preference for the alternative prey). However, empirical studies do not result in clear all-or-nothing responses. Previous models examined the relaxation of the all-or-nothing response by assuming partial preference (e.g., predators preferentially forage on the primary prey even when they also attack the alternative prey). In this study, I consider individual variation in two predator traits (prey density perception and handling time) as the sources of the variation in the threshold density, which can make empirical data appear deviated from the expectation. I examine how community models with partial preference and individual variation differ in their dynamics and show that the differences can be substantial. For example, the dynamics of a model based on individual variation can be more stable (e.g., stable in a wider parameter region) than that of a model based on partial preference. As the general statistical property (Jensen’s inequality) is a main factor that causes the differences, the results of the study have general implications to the interpretation of models based on average per-capita rates.     

Parameterising variable assimilation efficiency in predator–prey models

Our understanding of the dynamics of predator–prey systems has relied heavily on the use of models based on the standard Lotka–Volterra (LV) framework, dating back over 80 years. Although these models have been repeatedly analysed and refined since their initial inception, the way they describe the predator's growth rate has received surprisingly little attention; typically it is simply assumed that the predator's growth rate is linearly related to its ingestion rate according to a constant assimilation efficiency, e. However, for many consumers e is known to decrease at high prey densities. Models that ignore variable assimilation efficiencies overlook potentially important non‐linearities, affecting the validity of predictions relating to conservation, invasion biology and pest control. Directly quantifying the relationship between e and prey abundance is, however, difficult. An alternative approach (the independent‐response, IR, approach) is to not assume any direct link between the predator's functional response (the relationship between ingestion rate and prey abundance) and its growth response. This flexibility is invaluable when parameterising models from data; providing the model‐fitting process is constrained to ensure that e never exceeds 1, this approach allows considerable insight into whether, and how, e varies with prey density. Here we examine the synergistic value of combining the IR and LV approaches. We illustrate these concepts through analysis of published functional and growth response data and show that, in many cases, e does vary with prey abundance. This paper is the first recognition that these two complementary approaches can be combined into a single framework that allows the relationship between a predator's functional and growth responses to emerge during the parameterisation process, thereby acting as a compromise between restrictive models that require this relationship to be defined a priori, and completely unrestrained models that allow assimilation efficiencies to exceed 1.     

Explicit model for searching behavior of predator

The authors present an approach for explicit modeling of spatio-temporal dynamics of predator-prey community. This approach is based on a reaction-diffusion-adjection PD (prey dependent) system. Local kinetics of population is determined by logistic reproduction function of prey, constant natural mortality of predator and Holling type 2 trophic function. Searching behavior of predator is described by the advective term in predator balance equation assuming the predator acceleration to be proportional to the prey density gradient. The model was studied with zero-flux boundary conditions. The influence of predator searching activity on the community dynamics, in particular, on the emergence of spatial heterogeneity, has been investigated by linear analysis and numerical simulations. It has been shown how searching activity may effect the persistence of species, stabilizing predator-prey interactions at very low level of pest density. It has been demonstrated that obtaining of such dynamic regimes does not require the use of complex trophic functions.     

Artificial selection for timing of dispersal in predatory mites yields lines that differ in prey exploitation strategies

Dispersal is the main determinant of the dynamics and persistence of predator–prey metapopulations. When defining dispersal as a predator exploitation strategy, theory predicts the existence of a continuum of strategies: from some dispersal throughout the predator–prey interaction (the Milker strategy) to dispersal only after the prey had been exterminated (the Killer strategy). These dispersal strategies relate to differences in prey exploitation at the population level, with more dispersal leading to longer predator–prey interaction times and higher cumulative numbers of dispersing predators. In the predatory mite Phytoseiulus persimilis, empirical studies have shown genetic variation for prey exploitation as well as for the timing of aerial dispersal in the presence of prey. Here, we test whether artificial selection for lines that differ in timing of dispersal also results in these lines differing in prey exploitation. Six rounds of selection for early or late dispersal resulted in predator lines displaying earlier or later dispersal. Moreover, it resulted—at the population level—in predicted differences in the local predator–prey interaction time and in the cumulative numbers of dispersers in a population dynamics experiment. We pose that timing of dispersal is a heritable trait that can be selected in P. persimilis, which results in lines that show quantitative differences in local predator–prey dynamics. This opens ways to experimentally investigate the evolution of alternative prey exploitation strategies and to select for predator strains with prey exploitation strategies resulting in better biological control.     

Influence of predator‐specific defense adaptation on intraguild predation

The persistence of intraguild predation (IGP), the prey–predator interaction between competing species, is puzzling because simple IGP models readily predict species extinction. In this study, we explored a mathematical model incorporating predator‐specific defense adaptation of basal prey against intraguild prey and intraguild predator. The model explicitly described the dynamics of the defense effort against each predator under the assumption that anti‐predator defense was associated with reducing effort allocated to reproduction. The model predicted that defense adaptation (i.e. the ability to reallocate defense effort) would facilitate coexistence, particularly when system productivity is high; at low productivity, coexistence would be facilitated or inhibited depending on initial effort allocation prior to defense adaptation. In addition, we found that three‐species dynamics became more stable at higher adaptation rates. The results suggest that common behavioral changes, such as predator‐specific defense adaptation, have significant implications for the community structure and dynamics of IGP systems.     

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.     

Dynamics of a intraguild predation model with generalist or specialist predator

Intraguild predation (IGP) is a combination of competition and predation which is the most basic system in food webs that contains three species where two species that are involved in a predator/prey relationship are also competing for a shared resource or prey. We formulate two intraguild predation (IGP: resource, IG prey and IG predator) models: one has generalist predator while the other one has specialist predator. Both models have Holling-Type I functional response between resource-IG prey and resource-IG predator; Holling-Type III functional response between IG prey and IG predator. We provide sufficient conditions of the persistence and extinction of all possible scenarios for these two models, which give us a complete picture on their global dynamics. In addition, we show that both IGP models can have multiple interior equilibria under certain parameters range. These analytical results indicate that IGP model with generalist predator has “top down” regulation by comparing to IGP model with specialist predator. Our analysis and numerical simulations suggest that: (1) Both IGP models can have multiple attractors with complicated dynamical patterns; (2) Only IGP model with specialist predator can have both boundary attractor and interior attractor, i.e., whether the system has the extinction of one species or the coexistence of three species depending on initial conditions; (3) IGP model with generalist predator is prone to have coexistence of three species.     

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.     

Parametric Analysis of a Predator–prey System Stabilized by a Top Predator

We present a complete parametric analysis of a predator–prey system influenced by a top predator. We study ecosystems with abundant nutrient supply for the prey where the prey multiplication can be considered as proportional to its density. The main questions we examine are the following: (1) Can the top predator stabilize such a system at low densities of prey? (2) What possible dynamic behaviors can occur? (3) Under which conditions can the top predation result in the system stabilization? We use a system of two nonlinear ordinary differential equations with the density of the top predator as a parameter. The model is investigated with methods of qualitative theory of ODEs and the theory of bifurcations. The existence of 12 qualitatively different types of dynamics and complex structure of the parametric space are demonstrated. Our studies of phase portraits and parametric diagrams show that a top predator can be an important factor leading to stabilization of the predator-prey system with abundant nutrient supply. Although the model here is applied to the plankton communities with fish (or carnivorous zooplankton) as the top trophic level, the general form of the equations allows applications of our results to other ecological systems.     

Predator-prey interactions in a nonequilibrium context: the metapopulation approach to modeling "hide-and-seek" dynamics in a spatially explicit tri-trophic system

Predators and prey are usually heterogeneously distributed in space so that the ability of the predators to respond to the distribution of their prey may have a profound influence on the stability and persistence of a predator‐prey system. A special type of dynamics is “hide‐and‐seek” characterized by a high turnover rate of local populations of prey and predators, because once the predators have found a patch of prey they quickly overexploit it, whereupon the starving predators either should move to better places or die. Continued persistence of prey and predators thus hinges on a long‐term balance between local extinctions and founding of new subpopulations. The colonization rate depends on the rate of emigration from occupied patches and the likelihood of successfully arriving at a suitable new patch, while extinction rate depends on the local population dynamics. Since extinctions and colonizations are both discrete probabilistic events, these phenomena are most adequately modeled by means of a stochastic model. In order to demonstrate the qualitative differences between a deterministic and stochastic approach to population dynamics, a spatially explicit tritrophic predator‐prey model is developed in a deterministic and a stochastic version. The model is parameterized using data for the two‐spotted spider mite (Tetranychus urticae) and the phytoseiid mite predator Phytoseiulus persimilis inhabiting greenhouse cucumbers.
Simulations show that the deterministic and stochastic approaches yield different results. The deterministic version predicts that the populations will exhibit violent fluctuations, implying that the system is fundamentally unstable. In contrast, the stochastic version predicts that the two species will be able to coexist in spite of frequent local extinctions of both species, provided the system consists of a sufficiently large number of local populations. This finding is in agreement with experimental results. It is therefore concluded that demographic stochasticity in combination with dispersal is capable of producing and maintaining sufficient asynchrony between local populations to ensure long‐term regional (metapopulation) persistence.     

A synthetic Escherichia coli predator–prey ecosystem

We have constructed a synthetic ecosystem consisting of two Escherichia coli populations, which communicate bi‐directionally through quorum sensing and regulate each other's gene expression and survival via engineered gene circuits. Our synthetic ecosystem resembles canonical predator–prey systems in terms of logic and dynamics. The predator cells kill the prey by inducing expression of a killer protein in the prey, while the prey rescue the predators by eliciting expression of an antidote protein in the predator. Extinction, coexistence and oscillatory dynamics of the predator and prey populations are possible depending on the operating conditions as experimentally validated by long‐term culturing of the system in microchemostats. A simple mathematical model is developed to capture these system dynamics. Coherent interplay between experiments and mathematical analysis enables exploration of the dynamics of interacting populations in a predictable manner.