A mathematical model of population dynamics for Batesian mimicry system

We analyse a mathematical model of the population dynamics among a mimic, a corresponding model, and their common predator populations. Predator changes its search-and-attack probability by forming and losing its search image. It cannot distinguish the mimic from the model. Once a predator eats a model individual, it comes to omit both the model and the mimic species from its diet menu. If a predator eats a mimic individual, it comes to increase the search-and-attack probability for both model and mimic. The predator may lose the repulsive/attractive search image with a probability per day. By analysing our model, we can derive the mathematical condition for the persistence of model and mimic populations, and then get the result that the condition for the persistence of model population does not depend on the mimic population size, while the condition for the persistence of mimic population does depend the predator's memory of search image.     

Assessing the viability of tri-trophic food chain model in designing a conservation plan: The case of dwindling Quokka population

The extinction and persistence dynamics of quokka (Setonix brachyurus) population at northern jarrah forest of Australia is investigated using mathematical modelling. Predator's management demands a comprehensive understanding of the ecological circumstances associated with predation. Predation by red foxes (Vulpes vulpes) has a significant role in reducing the native animal population. This paper mainly focuses on the extinction dynamics of quokka population and its reduction by red foxes, by qualitative and quantitative analyses of a designed tri-trophic food-chain model composing a prey (quokka), a mesopredator (red fox) and apex predator (dingo). Existence of solution are analysed and shown to be uniformly bounded. We applied the concept of basic reproduction number from epidemiology to the food chain model, to derive a condition for extinction and persistence of predator population. Global stability of the predator-free equilibrium is established by geometric approach. We use Partial Rank Correlation Coefficient (PRCC) for performing global sensitivity analysis to identify most influential model parameter responsible for quokkas prevalence and mortality. Results of numerical simulation for both deterministic and stochastic model confirms the analytical finding and support those of previous studies. The outcome of this work shows that persistence and existence of quokka depend on the demographic impacts of environmental stochasticity on its own population. Based on our results, many conservation strategies are suggested to improve the overall growth of the species. We believe that declination of red fox and the presence of dingoes are important for preserving the uniform occurrence of quokkas.     

A population dynamic model of Batesian mimicry

A population dynamic model of Batesian mimicry, in which populations of both model and mimetic species were considered, was analyzed. The probability of a predator catching prey on each encouter was assumed to depend on the frequency of the mimic. The change in population size of each species was considered to have two components, growth at the intrinsic growth rate and carrying capacity, and reduction by predation. For simplicity in the analyses, three assumptions were made concerning the carrying capacities of each population: (1) with no density effects on the mimic population growth rate; (2) with no density effects on the model species; and (3) with density effects on both species. The first and second cases were solved analytically, whereas the last was, for the most part, investigated numerically. Under assumption (1), two stable equilibria are possible, in which both species either coexist or go to extinction. Under assumption (2), there are also two stable equilibria possible, in which either only the mimic persists or both go to extinction. These results explain the field records of butterflies (Pachliopta aristolochiae and its mimic Papilio polytes) in the Ryukyu Islands, Japan.     

Effects of the probability of a predator catching prey on predator–prey system stability

To understand the effect of the probability of a predator catching prey, P_catch, on the stability of the predator–prey system, a spatially explicit lattice model consisting of predators, prey, and grass was constructed. The predators and prey randomly move on the lattice space, and the grass grows according to its growth probability. When a predator encounters prey, the predator eats the prey in accordance with the probability P_catch. When a prey encounters grass, the prey eats the grass. The predator and prey give birth to offspring according to a birth probability after eating prey or grass, respectively. When a predator or prey is initially introduced or newly born, its health state is set at a high given value. This health state decreases by one with every time step. When the state of an animal decreases to less than zero, the individual dies and is removed from the system. Population densities for predator and prey fluctuated significantly according to P_catch. System stability was characterized by the standard deviation ? of the fluctuation. The simulation results showed that ? for predators increased with an increase of P_catch; ? for prey reached a maximum at P_catch = 0.4; and ? for grass fluctuated little regardless of P_catch. These results were due to the tradeoff between P_catch and the predator–prey encounter rate, which represents the degree of interaction between predator and prey and the average population density, respectively.     

The Role of Density Regulation in Extinction Processes and Population Viability Analysis

We review the role of density dependence in the stochastic extinction of populations and the role density dependence has played in population viability analysis (PVA) case studies. In total, 32 approaches have been used to model density regulation in theoretical or applied extinction models, 29 of them are mathematical functions of density dependence, and one approach uses empirical relationships between density and survival, reproduction, or growth rates. In addition, quasi-extinction levels are sometimes applied as a substitute for density dependence at low population size. Density dependence further has been modelled via explicit individual spacing behaviour and/or dispersal. We briefly summarise the features of density dependence available in standard PVA software, provide summary statistics about the use of density dependence in PVA case studies, and discuss the effects of density dependence on extinction probability. The introduction of an upper limit for population size has the effect that the probability of ultimate extinction becomes 1. Mean time to extinction increases with carrying capacity if populations start at high density, but carrying capacity often does not have any effect if populations start at low numbers. In contrast, the Allee effect is usually strong when populations start at low densities but has only a limited influence on persistence when populations start at high numbers. Contrary to previous opinions, other forms of density dependence may lead to increased or decreased persistence, depending on the type and strength of density dependence, the degree of environmental variability, and the growth rate. Furthermore, effects may be reversed for different quasi-extinction levels, making the use of arbitrary quasi-extinction levels problematic. Few systematic comparisons of the effects on persistence between different models of density dependence are available. These effects can be strikingly different among models. Our understanding of the effects of density dependence on extinction of metapopulations is rudimentary, but even opposite effects of density dependence can occur when metapopulations and single populations are contrasted. We argue that spatially explicit models hold particular promise for analysing the effects of density dependence on population viability provided a good knowledge of the biology of the species under consideration exists. Since the results of PVAs may critically depend on the way density dependence is modelled, combined efforts to advance statistical methods, field sampling, and modelling are urgently needed to elucidate the relationships between density, vital rates, and extinction probability.     

Batesian mimicry and signal detection theory

Signal Detection Theory can be used to provide a mathematical model describing the choice of a predator trying to distinguish between a model and a Batesian mimic. The mathematical model yields a number of a deductions, in particular that it may or may not assist the mimic population if mimics more closely resemble their models. The assumptions underlying the analysis are discussed in some detail.     

Obligate mutualism with a predator: Stability and persistence of three-species models

We present a general model for three interacting populations, where one population, called a mutualist, benefits a predator in its interaction with the prey. Biologically, there are four different ways in which the mutualist could benefit the predator: by enhancing prey growth rate, by enhancing the rate of prey capture, by providing an alternative food supply for the predator, and by enhancing the efficiency of utilization of prey, once they are ingested. We discuss examples of each type of interaction. We restrict our model to those situations in which the predator cannot survive on the prey in the absence of the mutualist. Therefore, if mutualism exists, it is obligate for the predator. Other conditions of the model include the dynamics of the prey and the mutualist alone and together in the absence of the predator. Given additional reasonable restrictions on the model, we determine the conditions for persistence, where persistence is defined as the continued existence of all three populations without any of them going extinct. There are two ways in which survival may arise in these models. Under one set of conditions, which is equivalent to the predator being able to invade a prey-mutualist system when rare, persistence will occur for any set of positive critical population sizes. Alternatively, survival will occur if there is an asymptotically stable interior equilibrium. However, the conditions for this are complex, and survival may occur only for initial populations in a limited region around the equilibrium.     

Cannibalism in discrete-time predator-prey systems

In this study, we propose and investigate a two-stage population model with cannibalism. It is shown that cannibalism can destabilize and lower the magnitude of the interior steady state. However, it is proved that cannibalism has no effect on the persistence of the population. Based on this model, we study two systems of predator-prey interactions where the prey population is cannibalistic. A sufficient condition based on the nontrivial boundary steady state for which both populations can coexist is derived. It is found via numerical simulations that introduction of the predator population may either stabilize or destabilize the prey dynamics, depending on cannibalism coefficients and other vital parameters.     

Evolutionary consequences of a search image

Many predators are able to become better at spotting cryptic prey by recognising specific clues, but by concentrating on one prey type they will become worse at spotting other prey types. This phenomenon is known as the formation of a search image for a certain prey by a predator and is related to apostatic selection. Here, we study the evolution of a search image in the predator by formulating and analysing a mathematical model. The predator forages for two prey types and is able to form an independent search image for both prey. The results show that the evolutionary dynamics can be divided into two parts: a fast and a slow part. At first selection pressure will be strong towards a stable ratio of prey, which is the same as the ratio found for the unbeatable prey choice for predators with a Holling type II functional response. Following this, the slow dynamics will keep this ratio constant independent of the trait values, but the predator will slowly evolve towards a stronger search image and ultimately become a specialist predator or slowly evolve towards generalist with a weak search image. In conclusion, the formation of a search image causes the predator to control the prey densities such that the ratio of available prey is kept constant by the predator.     

The effects of habitat fragmentation on persistence of source-sink metapopulations in systems with predators and prey or apparent competitors.

We consider systems with one predator and one prey, or a common predator and two prey species (apparent competitors) in source and sink habitats. In both models, the predator species is vulnerable to extinction, if productivity in the source is insufficient to rescue demographically deficient sink populations. Conversely, in the model with two prey species, if the source is too rich, one of the prey species may be driven extinct by apparent competition, since the predator can maintain a large population because of the alternative prey. Increasing the rate of predator movement from the source population has opposite effects on prey and predator persistence. High emigration rate exposes the predator population to danger of extinction, reducing the number of individuals that breed and produce offspring in the source habitat. This may promote coexistence of prey by relaxing predation pressure and apparent competition between the two prey species. The number of sinks and spatial arrangement of patches, or connectivity between patches, also influence persistence of the species. More sinks favor the prey and fewer sinks are advantageous to the predator. A linear pattern with the source at one end is profitable for the predator, and a centrifugal pattern in which the source is surrounded by sinks is advantageous to the prey. When the dispersal rate is low, effects of the spatial structure may exceed those of the number of sinks. In brief, productivity in patches and patterns of connectivity between patches differentially influence persistence of populations in different trophic levels.     

Diseased Social Predators

Social predators benefit from cooperation in the form of increased hunting success, but may be at higher risk of disease infection due to living in groups. Here, we use mathematical modeling to investigate the impact of disease transmission on the population dynamics benefits provided by group hunting. We consider a predator–prey model with foraging facilitation that can induce strong Allee effects in the predators. We extend this model by an infectious disease spreading horizontally and vertically in the predator population. The model is a system of three nonlinear differential equations. We analyze the equilibrium points and their stability as well as one- and two-parameter bifurcations. Our results show that weakly cooperating predators go unconditionally extinct for highly transmissible diseases. By contrast, if cooperation is strong enough, the social behavior mediates conditional predator persistence. The system is bistable, such that small predator populations are driven extinct by the disease or a lack of prey, and large predator populations survive because of their cooperation even though they would be doomed to extinction in the absence of group hunting. We identify a critical cooperation level that is needed to avoid the possibility of unconditional predator extinction. We also investigate how transmissibility and cooperation affect the stability of predator–prey dynamics. The introduction of parasites may be fatal for small populations of social predators that decline for other reasons. For invasive predators that cooperate strongly, biocontrol by releasing parasites alone may not be sufficient.     

Ecological stability and social hierarchy

We have examined a predator-prey model in which the predator is assumed to have a social structure of the dominance hierarchy or “peck order” type in which the feeding success of an individual is related both to the availability of food and to his social rank. We find such a social structure to be a strongly beneficial influence on population stability so long as the rewards of social dominance are not too extreme. We also show that an optimally hierarchical predator population can stably achieve a much larger depression of the prey below its carrying capacity than is possible for a simple predator population composed of identical individuals. This strongly suggests that socially structured predator populations may be more effective agents of biological control than simpler predators with no such population structure.     

Dynamic Model for Life History of Scyphozoa

A two-state life history model governed by ODEs is formulated to elucidate the population dynamics of jellyfish and to illuminate the triggering mechanism of its blooms. The polyp-medusa model admits trichotomous global dynamic scenarios: extinction, polyps survival only, and both survival. The population dynamics sensitively depend on several biotic and abiotic limiting factors such as substrate, temperature, and predation. The combination of temperature increase, substrate expansion, and predator diminishment acts synergistically to create a habitat that is more favorable for jellyfishes. Reducing artificial marine constructions, aiding predator populations, and directly controlling the jellyfish population would help to manage the jellyfish blooms. The theoretical analyses and numerical experiments yield several insights into the nature underlying the model and shed some new light on the general control strategy for jellyfish.     

Consequences of the Allee Effect and Intraspecific Competition on Population Persistence under Adverse Environmental Conditions

The impact of intraspecific interactions on ecological stability and population persistence in terms of steady state(s) existence is considered theoretically based on a general competition model. We compare persistence of a structured population consisting of a few interacting (competitive) subpopulations, or groups, to persistence of the corresponding unstructured population. For a general case, we show that if the intra-group competition is stronger than the inter-group competition, then the structured population is less prone to extinction, i.e. it can persist in a parameter range where the unstructured population goes extinct. For a more specific case of a population with hierarchical competition, we show that relative viability of structured and unstructured populations depend on the type of density dependence in the population growth. Namely, while in the case of logistic growth, structured and unstructured populations exhibit equivalent persistence; in the case of Allee dynamics, the persistence of a hierarchically structured population is shown to be higher. We then apply these results to the case of behaviourally structured populations and demonstrate that an extreme form of individual aggression can be beneficial at the population level and enhance population persistence.     

Dynamics of a stochastic predator-prey model with habitat complexity and prey aggregation

Interplay between predator and prey is a complex process in ecosystems due to its nature. The population dynamics can be affected by many extrinsic and intrinsic factors. In this paper, we make an attempt to uncover the effects from environmental disturbances when populations are subject to habitat complexity and aggregation effect. We firstly propose a stochastic predator-prey model with habitat complexity and aggregation efficiency for prey. We then mathematically analyze the model, to demonstrate the existence, uniqueness and the stochastically ultimately boundedness of the global positive solution, and to establish sufficient conditions for the existence of ergodic stationary distribution of the solution. We also establish sufficient conditions under which either only predator population dies out or the entire predator-prey model becomes extinct. Our theoretical and numerical results indicate that: (1) the environmental noises are disadvantage for the survival of biological populations; (2) when the density of prey is greater than one, prey aggregation can heighten the capability of predator species to capture prey and reduce the effect of environmental fluctuations, while when the density of prey is less than one, the results are opposite; (3) habitat complexity is propitious to the survival of prey population and may seriously threaten the persistence of the predator population.     

A model of predator-prey dynamics as modified by the action of a parasite

A predator-prey population is described in which the prey population may be either a secondary host or a primary host to a parasite, but the predator is always a primary host. Those prey that have been invaded by the parasite have their behavior modified so as to make them more susceptible to predation. The model is described by a system of three autonomous ordinary differential equations. Conditions for persistence of all populations are given in the case that both populations are primary hosts. A brief discussion of the stability of the interior equilibrium is given.     

Frequency-dependent predation and maintenance of prey polymorphism

In positive frequency-dependent predation, predation risk of an individual prey correlates positively with the frequency of that prey type. In a number of small-scale experiments individual predators have shown frequency-dependent behaviour, often leading to the conclusion that a population of such predators could maintain prey polymorphism. Using simulations, I studied the dynamics of frequency-dependent predation and prey polymorphism. The model suggests that persistence of prey polymorphism decreases with increasing number of predators that show frequency-dependent behaviour, questioning conclusions about polymorphism based on experiments with few predators. In addition, prey population size, prey crypsis, difference in crypsis between prey morphs and the way the behaviour was adjusted affected the persistence of polymorphism. Under some circumstances prey population remained polymorphic for a shorter time under frequency-dependent than under frequency-independent predation. This suggests that although positive frequency-dependent predator behaviour may maintain prey polymorphism, it is not a sufficient condition for persistent prey polymorphism.     

A stochastic foraging model with predator training effects. II. Optimal diets

Standard optimal diet models require that a predator's behavior while searching for food does not change in response to experiences with individual prey. There is evidence for rapid and reversible changes in feeding behavior caused by as few as one or two prey encounters. When these “training effects” occur, a given prey type is more likely to be captured next if it was the last type with which the predator had experience. This is not compatible with the standard foraging model. I present a stochastic model which incorporates predator training effects, and three types of training are explored: training in the ability to detect prey (search image formation), training in the probability of succeeding in an attempted capture, and training in the time to pursue, capture, and eat prey. The main result is that all three types of training can result in optimal diets which do not obey the standard optimal diet rules. Conditions under which these rules will suffice are discussed.     

The dynamics of multispecies, multi-life-stage models of aquatic food webs

We investigated the dynamics of models of aquatic food webs using stability analysis methods previously applied to other types of food web models. Our models expanded traditional Lotka-Volterra models of predator-prey interactions in several ways. We added life history structure to these models in order to investigate its effects. Life history omnivory is different life history stages of a species feeding in trophically different positions in a food web. Such a species might appear omnivorous, integrating across all stages, but the individual stage might not be. Other important additions to the basic models included stock-recruitment relationships between adults and young and food-dependent maturation rates for early life history stages. Complex models of multispecies interactions were built from basic ones by adding new features sequentially. Our analysis revealed five major features of our multispecies, multi-life-stage models. Omnivory reduces stability, as it does in food web models without life history structure. However, life history omnivory reduces stability much less than single life stage omnivory does. Stock recruitment relationships affect the likelihood of finding stable models. If the maturation rate of young varies with their food supply, the chance of finding stable models decreases. Finally, predation loops of the type A eats B, B eats A, or A eats B, B eats C, C eats A greatly reduce model stability. We present both biological and mathematical explanations for these findings. We also discuss their implications for management of marine resources.