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.     

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.     

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.     

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.     

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.     

On some Markov models of certain interacting populations

We give a stochastic foundation to the Volterra prey-predator population in the following case. We take Volterra's predator equations and let a free host birth and death process support the evolution of the predator population. The purpose of this article is to present a rigorous population sample path construction of this interacted predator process and study the properties of this interacted process. The constructions yields a strong Markov process. The existence of steady-state distribution for the interacted predator process means the existence of equilibrium population level. We find a necessary and sufficient condition for the existence of a steady-state distribution. Next we see that if the host process possesses a steady-state distribution, so does the interacted predator process and this distribution satisfies a difference equation. For special choices of the auto death and interaction parametersa andb of the predator, whenever the host process visits the particular statea ^*=a/b the predator takes rest (saturates) from its evolution. We find the probability of asymptotic saturating of the predator.     

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.     

Dynamics ofPanonychus ulmi andTyphlodromus pyri: factors contributing to persistence

We addressed the question of persistence of predator and prey in a biological control system by examining temporal patterns ofPanonychus ulmi (Koch) and its predator,Typhlodromus pyri Scheuten at two geographic locations and at two spatial scales. At the scale of an orchard, bothP. ulmi andT. pyri were persistent over the time frame of 6 years. At the scale of an individual tree,T. pyri appeared to be more persistent than its prey,P. ulmi. We used a simulation model of single populations ofP. ulmi andT. pyri to determine which of several aspects of the biology of the two species could contribute to such a pattern. Spatial incongruity between predator and prey was essential for persistence of both species. The generalist food habit ofT. pyri probably contributes to the persistence ofT. pyri on individual trees, and may cause occasional extinction ofP. ulmi at this spatial scale. The presence of alternate food is likely an essential element for successful biological control in this system. Cannibalism byT. pyri results in higher prey densities, that is, it is detrimental to the biological control ofP. ulmi, but has no effect on the relative persistence of the two species.     

Predator presence dramatically reduces copepod abundance through condition‐mediated non‐consumptive effects

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Newly emerged Batesian mimicry protects only unfamiliar prey

The evolution of Batesian mimicry was tested experimentally using avian predators. We investigated the effect of a search image on the protection effectiveness of a newly emerged Batesian mimic. The two groups of predators (adult great tits, Parus major) differed in prior experience with prey from which the mimic evolved. The Guyana spotted roach (Blaptica dubia) was used as a palatable prey from which the mimic emerged, and red firebug (Pyrrhocoris apterus) was used as a model. Optical signalization of the insect prey was modified by a paper sticker placed on its back. The cockroaches with the firebug pattern sticker were significantly better protected against tits with no prior experience with cockroaches. The protection of the firebug sticker was equally effective on cockroaches as it was on firebugs. The cockroaches with firebug stickers were not protected against attacks of tits, which were familiar with unmodified cockroaches better than cockroaches with a cockroach sticker. We suppose that pre-trained tits acquired the search image of a cockroach, which helped them to reveal the “fake” Batesian mimic. Such a constraint of Batesian mimicry effectiveness could substantially decrease the probability of evolution of pure Batesian mimic systems.     

A population model to simulate northern bobwhite population dynamics in southern Texas

Models are important tools that can help managers and researchers understand the population dynamics of a species and how different habitat or population management scenarios impact that species. We used radio-telemetry data from northern bobwhites (Colinus virginianus) in southern Texas from 2000 to 2005 to develop a stochastic simulation model for bobwhite populations. Our model is based on difference equations, with stochastic variables drawn from normal and Weibull distributions. We simulated bobwhite populations to 100 yr and evaluated our model by comparing results with independent estimates of 4 population parameters (spring and fall density, finite rate of increase in the fall population [λ], and winter juv:ad age ratios). Using a quasi-extinction criterion of ≤40 birds (density = ≤0.05 birds/ha), probability of persistence to 100 yr was 88.3% (106 of 120 simulations) for the spring population and 96.7% (116 of 120 simulations) for the fall population. Using a less restrictive quasi-extinction criteria (≤14 birds), probability of persistence was 93.3% (112 of 120 simulations) for the spring population and 98.3% (118 of 120 simulations) for the fall population. Simulated population parameters were similar to independent estimates for 4 of 4 population parameters. Winter age ratios differed between our model ( juv:ad, n = 120, SE = 0.32) and empirical age ratios from harvested bobwhites on our study area ( juv:ad, n = 25, SE = 0.24). However, when we corrected harvest age ratios for bias in juvenile harvest ( juv:ad, n = 25, SE = 0.32) simulated and empirical estimates were similar. Our model appears to be a reliable predictor of bobwhite populations in the southern Texas. Our simulation results indicate that bobwhite hunters and managers can expect excellent bobwhite hunting (fall populations ≥2.2 birds per ha) in about one of 10 yr. © 2011 The Wildlife Society     

The Effects of Prey Patchiness, Predator Aggregation, and Mutual Interference on the Functional Response of Phytoseiulus persimilis Feeding on Tetranychus urticae (Acari: Phytoseiidae, Tetranychidae)

The spatial distributions of two-spotted spider mites Tetranychus urticae and their natural enemy, the phytoseiid predator Phytoseiulus persimilis, were studied on six full-grown cucumber plants. Both mite species were very patchily distributed and P. persimilis tended to aggregate on leaves with abundant prey. The effects of non-homogenous distributions and degree of spatial overlap between prey and predators on the per capita predation rate were studied by means of a stage-specific predation model that averages the predation rates over all the local populations inhabiting the individual leaves. The empirical predation rates were compared with predictions assuming random predator search and/or an even distribution of prey. The analysis clearly shows that the ability of the predators to search non-randomly increases their predation rate. On the other hand, the prey may gain if it adopts a more even distribution when its density is low and a more patchy distribution when density increases. Mutual interference between searching predators reduces the predation rate, but the effect is negligible. The stage-specific functional response model was compared with two simpler models without explicit stage structure. Both unstructured models yielded predictions that were quite similar to those of the stage-structured model.     

Large‐scale predator control increases population viability of a rare New Zealand riverine duck

The introduction of mammalian predators to oceanic islands has led to dramatic declines in the abundance of many native species. Conservation management of these species often relies on low‐cost predator control techniques that can be implemented over large scales. Assessing the effectiveness of such management techniques is difficult, but using population viability analyses (PVA), which identify the population growth rate (λ) and extinction risk of threatened species, may offer a solution. PVA provide the opportunity to compare the relative effectiveness of various management options and can identify knowledge gaps to prioritize research efforts. We used PVA to assess the population viability of whio (Hymenolaimus malacorhynchos), a rare riverine duck endemic to New Zealand. Current populations are threatened by introduced mammalian predators and are rapidly declining in both distribution and abundance. Whio conservation management is dominated by large‐scale, low‐intensity predator control, targeting introduced stoats (Mustela erminea). There is evidence that such control increases whio productivity but it is unknown if this increase is sufficient for long‐term population persistence. We undertook a stochastic PVA to assess the viability of whio populations under different management scenarios using data obtained from a 6‐year study of whio demographic responses to predator control. Populations with no predator control and low productivity will rapidly decline to extinction. Increasing productivity through predator control increased population viability but populations still showed a declining trajectory. A perturbation analysis showed that the growth rate of whio populations was largely driven by adult survival. Therefore, future research should target obtaining more robust estimates of adult survival, particularly how it is affected by predator control. Overall, our analysis indicated that large‐scale predator control increases the short‐term viability of whio populations but is insufficient for long‐term population persistence.     

A note on persistence about structured population models

In this paper, we report some results on persistence in two structured population models: a chronic- age-structured epidemic model and an age-duration-structured epidemic model. Regarding these models, we observe that the system is uniformly strongly persistent, which means, roughly speaking, that the proportion of infected subpopulation is bounded away from 0 and the bound does not depend on the initial data after a sufficient long time, if the basic reproduction ratio is larger than one. We derive this by adopting Thieme's technique, which requires some conditions about positivity and compactness. Although the compactness condition is rather difficult to show in general infinite-dimensional function spaces, we can apply Fréchet–Kolmogorov L ¹-compactness criteria to our models. The two examples that we study illuminate a useful method to show persistence in structured population models.     

Using population viability analysis,genomics, and habitat suitability to forecast future population patterns of Little Owl Athene noctua across Europe

The agricultural scene has changed over the past decades, resulting in a declining population trend in many species. It is therefore important to determine the factors that the individual species depend on in order to understand their decline. The landscape changes have also resulted in habitat fragmentation, turning once continuous populations into metapopulations. It is thus increasingly important to estimate both the number of individuals it takes to create a genetically viable population and the population trend. Here, population viability analysis and habitat suitability modeling were used to estimate population viability and future prospects across Europe of the Little Owl Athene noctua, a widespread species associated with agricultural landscapes. The results show a high risk of population declines over the coming 100 years, especially toward the north of Europe, whereas populations toward the southeastern part of Europe have a greater probability of persistence. In order to be considered genetically viable, individual populations must count 1,000–30,000 individuals. As Little Owl populations of several countries count <30,000, and many isolated populations in northern Europe count <1,000 individuals, management actions resulting in exchange of individuals between populations or even countries are probably necessary to prevent losing <1% genetic diversity over a 100‐year period. At a continental scale, a habitat suitability analysis suggested Little Owl to be affected positively by increasing temperatures and urban areas, whereas an increased tree cover, an increasing annual rainfall, grassland, and sparsely vegetated areas affect the presence of the owl negatively. However, the low predictive power of the habitat suitability model suggests that habitat suitability might be better explained at a smaller scale.     

Fungi,feather damage,and risk of predation

Predation is a powerful selective force with important effects on behavior, morphology, life history, and evolution of prey. Parasites may change body condition, health status, and ability to escape from or defend prey against predators. Once a prey individual has been detected, it can rely on a diversity of means of escape from the pursuit by the predator. Here we tested whether prey of a common raptor differed in terms of fungi from nonprey recorded at the same sites using the goshawk Accipiter gentilis and its avian prey as a model system. We found a positive association between the probability of falling prey to the raptor and the presence and the abundance of fungi. Birds with a specific composition of the community of fungi had higher probability of falling prey to a goshawk than individual hosts with fewer fungi. These findings imply that fungi may play a significant role in predator–prey interactions. The probability of having damaged feathers increased with the number of fungal colonies, and in particular the abundance of Myceliophthora verrucos and Schizophyllum sp. was positively related to the probability of having damaged feathers. In addition, we found a significant correlation between the rate of feather growth of goshawk prey with birds with more fungi being more likely to be depredated. These findings are consistent with the hypothesis that survival and feather quality of birds are related to abundance and diversity of fungi.     

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.     

Multiplex PCR amplification of 13 microsatellite loci for <Emphasis Type="Italic">Aquila chrysaetos</Emphasis> in forensic applications

The golden eagle (Aquila chrysaetos) is an endangered raptor, which is threatened mainly by illegal egg and nestling robbery. Here we describe a fluorescently labeled, multiplex PCR method using 13 microsatellite markers, which provides a powerful tool for the individual identification and parentage testing of the Golden eagle. This test should be applicable to both forensic analysis and population studies. Fifteen polymorphic loci from A. chrysaetos were cross-amplified. Subsequent PCR condition optimization led to the successful co-amplification of 13 different loci in a single PCR reaction. Fifty samples from wild-living individuals and 89 samples from captive-bred individuals were examined. The results indicated that both populations have similar levels of moderate inbreeding, unsurprising in a small population. This probability of excluding a random individual in parentage analysis was 0.9912 for the wild population and 0.9932 in the captive-bred one in the case that both the individual and its mother were examined together. The probability of identity was estimated to be 3 × 10⁻⁸ for the wild and 4 × 10⁻⁸ for the captive-bred populations. Given the size of the Slovak golden eagle population, this test should therefore be sufficient to reliably identify individual raptors and assess parentage in both conservation studies and forensic analysis.     

How the aphids got their spots: predation drives self‐organization of aphid colonies in a patchy habitat

Using a 30 day time series of aphid Aphis helianthi and coccinellid counts on 107 mapped racemes of Yucca glauca, we demonstrate progressive, predation‐induced self‐organization of aphid colonies on individual racemes into extremely low and extremely high population sizes. This was driven by a two‐attractor structure of density dependence that developed only in the presence of coccinellid predators. Foraging movements of the coccinellids among plants produced a power law relationship (average power = 0.142) between aphid and coccinellid numbers. This resulted in increased predation pressure on mid‐size colonies and decreased predation pressure on small and large populations. A field‐parameterized mathematical model predicts a two‐attractor structure in broad agreement with our observations. The overall system was integrated by the influence of the largest aphid populations, which determined the total number of coccinellids present, and thus the predation pressure throughout the system. Our study provides clear evidence of predator‐driven self‐organization of prey populations in a patchy environment.     

A predator prey model with age structure

A general predator-prey model is considered in which the predator population is assumed to have an age structure which significantly affects its fecundity. The model equations are derived from the general McKendrick equations for age structured populations. The existence, stability and destabilization of equilibria are studied as they depend on the prey's natural carrying capacity and the maturation periodm of the predator. The main result of the paper is that for a broad class of maturation functions positive equilibria are either unstable for smallm or are destabilized asm decreases to zero. This is in contrast to the usual rule of thumb that increasing (not decreasing) delays in growth rate responses cause instabilities.Research supported by National Science Foundation Grant No. MCS-7901307-01Research supported by National Scholarship for Study Abroad No. EDN/S-59/80 from the government of India