An incubating diseased-predator ecoepidemic model

We present a model for transmissible diseases spreading among predators in a predator–prey system. Upon successful contact, a susceptible individual becomes infected but is not yet able to spread the disease further. After an incubation period, the diseased individual becomes infectious. We investigate the system’s equilibria by analytical and numerical means. For a suitable set of parameter values, the system shows persistent oscillations. The model also exhibits bistability of the coexistence equilibrium with the prey-only equilibrium.     

Stochastic predation exposes prey to predator pits and local extinction

Understanding how predators affect prey populations is a fundamental goal for ecologists and wildlife managers. A well-known example of regulation by predators is the predator pit, where two alternative stable states exist and prey can be held at a low density equilibrium by predation if they are unable to pass the threshold needed to attain a high density equilibrium. While empirical evidence for predator pits exists, deterministic models of predator–prey dynamics with realistic parameters suggest they should not occur in these systems. Because stochasticity can fundamentally change the dynamics of deterministic models, we investigated if incorporating stochasticity in predation rates would change the dynamics of deterministic models and allow predator pits to emerge. Based on realistic parameters from an elk–wolf system, we found predator pits were predicted only when stochasticity was included in the model. Predator pits emerged in systems with highly stochastic predation and high carrying capacities, but as carrying capacity decreased, low density equilibria with a high likelihood of extinction became more prevalent. We found that incorporating stochasticity is essential to fully understand alternative stable states in ecological systems, and due to the interaction between top–down and bottom–up effects on prey populations, habitat management and predator control could help prey to be resilient to predation stochasticity.     

The role of size-specific predation in the evolution and diversification of prey life histories

Some of the best empirical examples of life-history evolution involve responses to predation. Nevertheless, most life-history theory dealing with responses to predation has not been formulated within an explicit dynamic food-web context. In particular, most previous theory does not explicitly consider the coupled population dynamics of the focal species and its predators and resources. Here we present a model of life-history evolution that explores the evolutionary consequences of size-specific predation on small individuals when there is a trade-off between growth and reproduction. The model explicitly describes the population dynamics of a predator, the prey of interest, and its resource. The selective forces that cause life-history evolution in the prey species emerge from the ecological interactions embodied by this model and can involve important elements of frequency dependence. Our results demonstrate that the strength of the coupling between predator and prey in the community determines many aspects of life-history evolution. If the coupling is weak (as is implicitly assumed in many previous models), differences in resource productivity have no effect on the nature of life-history evolution. A single life-history strategy is favored that minimizes the equilibrium resource density (if possible). If the coupling is strong, then higher resource productivities select for faster growth into the predation size refuge. Moreover, under strong coupling it is also possible for natural selection to favor an evolutionary diversification of life histories, possibly resulting in two coexisting species with divergent life-history strategies.     

Impacts of predation on dynamics of age-structured prey: Allee effects and multi-stability

With a series of mathematical models, we explore impacts of predation on a prey population structured into two age classes, juveniles and adults, assuming generalist, age-specific predators. Predation on any age class is either absent, or represented by types II or III functional responses, in various combinations. We look for Allee effects or more generally for multiple stable steady states in the prey population. One of our key findings is the occurrence of a predator pit (low-density ??refuge?? state of prey induced by predation; the chance of escaping predation thus increases both below and above an intermediate prey density) when only one age class is consumed and predators use a type II functional response ??this scenario is known to occur for an unstructured prey consumed via a type III functional response and can never occur for an unstructured prey consumed via a type II one. In the case where both age classes are consumed by type II generalist predators, an Allee effect occurs frequently, but some parameters give also rise to a predator pit and even three stable equilibria (one extinction equilibrium and two positive ones??Allee effect and predator pit combined). Multiple positive stable equilibria are common if one age class is consumed via a type II functional response and the other via a type III functional response??here, in addition to the behaviours mentioned above one may even observe three stable positive equilibria????double?? predator pit. Some of these results are discussed from the perspective of population management.     

Impact of fear effect on the growth of prey in a predator-prey interaction model

Several field data and experiments on a terrestrial vertebrates exhibited that the fear of predators would cause a substantial variability of prey demography. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. Based on the experimental evidence, we proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling type-II functional response. We investigate all the biologically feasible equilibrium points, and their stability is analyzed in terms of the model parameters. Our mathematical analysis exhibits that for strong anti-predator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. Our model system undergoes Hopf bifurcation by considering the birth rate r₀ as a bifurcation parameter. For larger prey birth rate, we investigate the transition to a stable coexisting equilibrium state, with oscillatory approach to this equilibrium state, indicating that the greatest characteristic eigenvalues are actually a pair of imaginary eigenvalues with real part negative, which is increasing for r₀. We obtained the conditions for the occurrence of Hopf bifurcation and conditions governing the direction of Hopf bifurcation, which imply that the prey birth rate will not only influence the occurrence of Hopf bifurcation but also alter the direction of Hopf bifurcation. We identify the parameter regions associated with the extinct equilibria, predator-free equilibria and coexisting equilibria with respect to prey birth rate, predator mortality rates. Fear can stabilize the predator-prey system at an interior steady state, where all the species can exists together, or it can create the oscillatory coexistence of all the populations. We performed some numerical simulations to investigate the relationship between the effects of fear and other biologically related parameters (including growth/decay rate of prey/predator), which exhibit the impact that fear can have in prey-predator system. Our numerical illustrations also demonstrate that the prey become less sensitive to perceive the risk of predation with increasing prey growth rate or increasing predators decay rate.     

Bistability and an Allee effect as emergent consequences of stage-specific predation

The Allee effect, a reduction of individual fitness at low population density that can lead to sudden and unannounced extinctions, has been shown to come about through a number of mechanisms, usually associated with group behavior or mate search. Recent papers show that it may arise through size-selective predation, without explicit assumptions relating individual fitness to population density. It arises from the shift that a predator induces in the population stage distribution of its prey. We study the parameter conditions that lead to such an emergent Allee effect. The emergent Allee effect occurs under fairly broad conditions. We show that stage-specific predation can also induce bistability between alternative states where both prey and predator are present. A perturbation analysis on the equilibria shows that all equilibria are highly robust to changes in predator density. Our work shows that when size-specific interactions are taken into account, bistabilities and catastrophic collapses are possible even in purely exploitative food webs, which has substantial implications for questions related to food web theory and conservation issues.     

High reproductive rates result in high predation risks: a mechanism promoting the coexistence of competing prey in spatially structured populations

I tested the hypothesis that spatial structure provides a trade-off between reproduction and predation risk and thereby facilitates predator-mediated coexistence of competing prey species. I compared a cellular automata model to a mean-field model of two prey species and their common predator. In the mean-field model, the prey species with the higher reproductive rate (the superior competitor) always outcompeted the other species (the inferior competitor), both in the presence of and the absence of the predator. In the cellular automata model, both prey species, which differed only in their reproductive rates, coexisted for a long time in the presence of their common predator at intermediate levels of predation. At low predation rates, the superior competitor dominated, while high predation rates favored the inferior competitor. This discrepancy in the results of the different models was due to a trade-off that spontaneously emerged in spatially structured populations; that is, the more clustered distribution of the superior competitor made it more susceptible to predation. In addition, coexistence of competing prey species declined with increasing dispersal ranges of either prey or predator, which suggests that the trade-off that results from spatial structure becomes less important as either prey or predator disperse over a broader range.     

Two-patch population models with adaptive dispersal: the effects of varying dispersal speeds

The population-dispersal dynamics for predator–prey interactions and two competing species in a two patch environment are studied. It is assumed that both species (i.e., either predators and their prey, or the two competing species) are mobile and their dispersal between patches is directed to the higher fitness patch. It is proved that such dispersal, irrespectively of its speed, cannot destabilize a locally stable predator–prey population equilibrium that corresponds to no movement at all. In the case of two competing species, dispersal can destabilize population equilibrium. Conditions are given when this cannot happen, including the case of identical patches.     

Body size distribution in <Emphasis Type="Italic">Daphnia</Emphasis> populations as an effect of prey selectivity by planktivorous fish

We test the hypothesis that size distribution of a Daphnia population reflects the vulnerability of each size category (instar) to predation by planktivorous fish. We hypothesize that due to the different reaction distances from which separate prey categories can be seen by a foraging fish, each category is preyed upon until its density is reduced and its size-specific apparent density level (number of prey within a hemisphere of radius equal to the reaction distance) or encounter rate (number of prey encountered per time within a tube with a cross section of radius equal to the reaction distance) become equal to those of other size categories. An experiment was performed with populations of Daphnia hyalina and D. pulicaria grown at two Scenedesmus + Chlamydomonas food levels (0.2 and 0.05 mg C per liter) in outdoor mesocosms (1000 l tanks) with predation by invertebrates (phantom midge) prevented by mosquito netting. Once the populations had become established, roach were added to the tanks at dusk each day and allowed to feed for 3 h, while control tanks were kept fish-free. After 20–60 days, while D. pulicaria was at low density level, the densities of D. hyalina in fish tanks were high enough to see that the age structure and size distribution matched those from simulations with the age-structured population model based on size-specific encounter rate. This match, however, remained only up to the point of first reproduction when—in contrast to the size/age distribution predicted by the model—the percentage share of adult instars in the total population decreased rapidly with age. This deviation from the predicted densities of adult instars suggests that neither encounter rates nor apparent densities derived from instar-specific reaction distances are sufficient to explain the instar-specific impact of a visual predator on planktonic prey. This implies that a foraging fish may temporarily change its feeding mode from the typical low-speed harvesting of small but abundant prey from within its visual field volume, to high-speed hunting for more scarce but larger ovigerous females when their abundance allows higher net energy gain. Shifting from one feeding mode to the other may be responsible for damping population density oscillations in Daphnia.     

Modeling the Fear Effect in Predator–Prey Interactions with Adaptive Avoidance of Predators

Recent field experiments on vertebrates showed that the mere presence of a predator would cause a dramatic change of prey demography. Fear of predators increases the survival probability of prey, but leads to a cost of prey reproduction. Based on the experimental findings, we propose a predator–prey model with the cost of fear and adaptive avoidance of predators. Mathematical analyses show that the fear effect can interplay with maturation delay between juvenile prey and adult prey in determining the long-term population dynamics. A positive equilibrium may lose stability with an intermediate value of delay and regain stability if the delay is large. Numerical simulations show that both strong adaptation of adult prey and the large cost of fear have destabilizing effect while large population of predators has a stabilizing effect on the predator–prey interactions. Numerical simulations also imply that adult prey demonstrates stronger anti-predator behaviors if the population of predators is larger and shows weaker anti-predator behaviors if the cost of fear is larger.     

Predation effects on the evolution of life-history traits in a clonal oligochaete

Although size at maturity and size and number of offspring are life-history traits widely studied in sexual and parthenogenetic reproduction, there is no such research on animals reproducing asexually without the involvement of gametes. Here we present an individual-based model in combination with experiments to study the clonal growth of Stylaria lacustris, an oligochaete reproducing through fission. We studied the effect of individual size at fission and fission ratio on clone fitness. Our results show that in benign environments without predators, fitness is higher when small worms produce small offspring. Then we included size-specific sublethal predation and found that the fitness of the clone is maximized when parental worms start fission at a large size and produce large descendants intercalated in the middle of the parental worm's body. These results agree with empirical findings. Furthermore, the results of our own laboratory experiment revealed that when S. lacustris is exposed to chemical alarm signals from injured conspecifics, it alters its life history in the same direction as predicted by the model. Our findings suggest that the effect of size-specific sublethal predation is similar to the effect of size-specific lethal predation because both modes of predation result in size-dependent prey mortality.     

Optimal predator management for mountain sheep conservation depends on the strength of mesopredator release

Large predators often suppress ungulate population growth, but they may also suppress the abundance of smaller predators that prey on neonatal ungulates. Antagonistic interactions among predators may therefore need to be integrated into predator–prey models to effectively manage ungulate–predator systems. We present a modeling framework that examines the net impact of interacting predators on the population growth rate of shared prey, using interactions among wolves Canis lupus, coyotes Canis latrans and Dall sheep Ovis dalli dalli as a case study. Wolf control is currently employed on approximately 16 million ha in Alaska to increase the abundance of ungulates for human harvest. We hypothesized that the positive effects of wolf control on Dall sheep population growth could be counteracted by increased levels of predation by coyotes. Coyotes and Dall sheep adult females (ewes) and lambs were radiocollared in the Alaska Range from 1999–2005 to estimate fecundity, age‐specific survival rates, and causes of mortality in an area without wolf control. We used stage‐structured population models to simulate the net effect of wolf control on Dall sheep population growth (λ). Our models accounted for stage‐specific predation rates by wolves and coyotes, compensatory mortality, and the potential release of coyote populations due to wolf control. Wolves were the main predators of ewes, coyotes were the main predators of lambs, and wolves were the main source of mortality for coyotes. Population models predicted that wolf control could increase sheep λ by 4% per year in the absence of mesopredator release. However, if wolf control released coyote populations, our models predicted that sheep λ could decrease by up to 3% per year. These results highlight the importance of integrating antagonistic interactions among predators into predator–prey models, because the net effect of predator management on shared prey can depend critically on the strength of mesopredator release.     

Backward bifurcation and oscillations in a nested immuno-eco-epidemiological model

This paper introduces a novel partial differential equation immuno-eco-epidemiological model of competition in which one species is affected by a disease while another can compete with it directly and by lowering the first species' immune response to the infection, a mode of competition termed stress-induced competition. When the disease is chronic, and the within-host dynamics are rapid, we reduce the partial differential equation model (PDE) to a three-dimensional ordinary differential equation (ODE) model. The ODE model exhibits backward bifurcation and sustained oscillations caused by the stress-induced competition. Furthermore, the ODE model, although not a special case of the PDE model, is useful for detecting backward bifurcation and oscillations in the PDE model. Backward bifurcation related to stress-induced competition allows the second species to persist for values of its invasion number below one. Furthermore, stress-induced competition leads to destabilization of the coexistence equilibrium and sustained oscillations in the PDE model. We suggest that complex systems such as this one may be studied by appropriately designed simple ODE models.     

Density-dependent effects of multiple predators sharing a common prey in an endophytic habitat

Multiple predator species feeding on a common prey can lead to higher or lower predation than would be expected by simply combining their individual effects. Such emergent multiple predator effects may be especially prevalent if predators share feeding habitat. Despite the prevalence of endophagous insects, no studies have examined how multiple predators sharing an endophytic habitat affect prey or predator reproduction. We investigated density-dependent predation of Thanasimus dubius (Coleoptera: Cleridae) and Platysoma cylindrica (Coleoptera: Histeridae) on a bark beetle prey, Ips pini (Coleoptera: Scolytidae), in a laboratory assay. I. pini utilize aggregation pheromones to group-colonize and reproduce within the stems of conifers. T. dubius and P. cylindrica exploit these aggregation pheromones to arrive simultaneously with the herbivore. Adult T. dubius prey exophytically, while P. cylindrica adults enter and prey within the bark beetle galleries. Larvae of both predators prey endophytically. We used a multiple regression analysis, which avoids confounding predator composition with density, to examine the effects of varying predator densities alone and in combination on herbivore establishment, herbivore reproduction, and predator reproduction. Predators reduced colonization success by both sexes, and decreased I. pini reproduction on a per male and per female basis. The combined effects of these predators did not enhance or reduce prey establishment or reproduction in unexpected manners, and these predators were entirely substitutable. The herbivores net replacement rate was never reduced significantly below one at prey and predator densities emulating field conditions. Similar numbers of each predator species emerged from the logs, but predator reproduction suffered from high intraspecific interference. The net replacement rate of P. cylindrica was not affected by conspecifics or T. dubius. In contrast, the net replacement rate of T. dubius decreased with the presence of conspecifics or P. cylindrica. Combinations of both predators led to an emergent effect, a slightly increased net replacement rate of T. dubius. This may have been due to predation by larval T. dubius on pupal P. cylindrica, as P. cylindrica develops more rapidly than T. dubius within this shared habitat.     

Differences in predator composition alter the direction of structure‐mediated predation risk in macrophyte communities

Structural complexity strongly influences the outcome of predator–prey interactions in benthic marine communities affecting both prey concealment and predator hunting efficacy. How habitat structure interacts with species‐specific differences in predatory style and antipredatory strategies may therefore be critical in determining higher trophic functions. We examined the role of structural complexity in mediating predator–prey interactions across several macrophyte habitats along a gradient of structural complexity in three different bioregions: western Mediterranean Sea (WMS), eastern Indian Ocean (EIO) and northern Gulf of Mexico (NGM). Using sea urchins as model prey, we measured survival rates of small (juveniles) and medium (young adults) size classes in different habitat zones: within the macrophyte habitat, along the edge and in bare sandy spaces. At each site we also measured structural variables and predator abundance. Generalised linear models identified biomass and predatory fish abundance as the main determinants of predation intensity but the efficiency of predation was also influenced by urchin size class. Interestingly though, the direction of structure‐mediated effects on predation risk was markedly different between habitats and bioregions. In WMS and NGM, where predation by roving fish was relatively high, structure served as a critical prey refuge, particularly for juvenile urchins. In contrast, in EIO, where roving fish predation was low, predation was generally higher inside structurally complex environments where sea stars were responsible for much of the predation. Larger prey were generally less affected by predation in all habitats, probably due to the absence of large predators. Overall, our results indicate that, while the structural complexity of habitats is critical in mediating predator–prey interactions, the direction of this mediation is strongly influenced by differences in predator composition. Whether the regional pool of predators is dominated by visual roving species or chemotactic benthic predators may determine if structure dampens or enhances the influence of top–down control in marine macrophyte communities.     

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.     

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.     

Harvesting and predation of a sex- and age-structured population

We study a discrete-time system of equations for a structured ungulate population exploited by human harvesting or a dynamic predator. The population is divided into juveniles, and female and male adults. Harvesting is concentrated on adults (trophy hunting of males or population control measures on females), whereas predation occurs in juveniles. Though the model consists of four nonlinear equations, we find explicit expressions for the steady states. We use these explicit expressions to investigate harvesting rates that allow population persistence, rates that ensure population control, and optimal harvesting efforts. Several reductions of complexity allow for a detailed analysis of the dynamics of the model. Most notably, we find that even compensatory density dependence can lead to a period doubling bifurcation, that the model does not support consumer–resource cycles, and that an Allee effect can emerge from the interplay of stage-specific predation and density-dependent prey reproduction.     

The complex dynamics of a diffusive prey–predator model with an Allee effect in prey

This paper investigates complex dynamics of a predator–prey interaction model that incorporates: (a) an Allee effect in prey; (b) the Michaelis–Menten type functional response between prey and predator; and (c) diffusion in both prey and predator. We provide rigorous mathematical results of the proposed model including: (1) the stability of non-negative constant steady states; (2) sufficient conditions that lead to Hopf/Turing bifurcations; (3) a prior estimates of positive steady states; (4) the non-existence and existence of non-constant positive steady states when the model is under zero-flux boundary condition. We also perform completed analysis of the corresponding ODE model to obtain a better understanding on effects of diffusion on the stability. Our analytical results show that the small values of the ratio of the prey's diffusion rate to the predator's diffusion rate are more likely to destabilize the system, thus generate Hopf-bifurcation and Turing instability that can lead to different spatial patterns. Through numerical simulations, we observe that our model, with or without Allee effect, can exhibit extremely rich pattern formations that include but not limit to strips, spotted patterns, symmetric patterns. In addition, the strength of Allee effects also plays an important role in generating distinct spatial patterns.     

Optimal harvesting of prey–predator system with interval biological parameters: A bioeconomic model

The paper presents the study of one prey one predator harvesting model with imprecise biological parameters. Due to the lack of precise numerical information of the biological parameters such as prey population growth rate, predator population decay rate and predation coefficients, we consider the model with imprecise data as form of an interval in nature. Many authors have studied prey–predator harvesting model in different form, here we consider a simple prey–predator model under impreciseness and introduce parametric functional form of an interval and then study the model. We identify the equilibrium points of the model and discuss their stabilities. The existence of bionomic equilibrium of the model is discussed. We study the optimal harvest policy and obtain the solution in the interior equilibrium using Pontryagin’s maximum principle. Numerical examples are presented to support the proposed model.