Dynamical Adaptation of Parental Care

Two models are proposed to simulate population growth of species with mature stage and immature stage in which there are parental cares for immature. It is assumed that the protection of mature to their immature reduces mortality of immature at the cost of reduction of reproduction. Dynamical adaptation of parental care is incorporated into the models, one of which is described with the proportional transition rate from immature to mature (ODE model) and the other one is described with a transition rate from immature to mature according to a fixed age (DDE model). For the ODE model, it is shown that the adaptation of parental care enlarges the possibility of species survival in the sense that population is permanent under the influences of the adaptation, but becomes extinct in the absence of adaptation. It is proved that the outcome of the adaptation makes the population in an optimal state. It is also observed that there are parental care switches, from noncare strategy to care strategy, as the natural death rate of immature individuals increases. The analysis of the DDE model indicates that the adaptation also enlarges the opportunity of population persistence, but the stage delay has the tendency to hinder the movement of population evolution to the optimal state. It is found that the loss rate of immature in the absence of parental care can induce different patterns to disturb the adaptation of population to optimal state. However, it is shown that the adaptation of parental care approaches to the optimal state when parental care is required for the survival of the population, for example, when the loss rate of immature or competition among mature increases or the fecundity decreases. The research was supported by Heiwa–Nakajima Fund and National Science Fund of China (No. 10571143). The research was partly supported by the Sasakawa Scientific Research Grant from The Japan Science Society. The research was supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.     

Disease transmission models with density-dependent demographics

The models considered for the spread of an infectious disease in a population are of SIRS or SIS type with a standard incidence expression. The varying population size is described by a modification of the logistic differential equation which includes a term for disease-related deaths. The models have density-dependent restricted growth due to a decreasing birth rate and an increasing death rate as the population size increases towards its carrying capacity. Thresholds, equilibria and stability are determined for the systems of ordinary differential equations for each model. The persistence of the infectious disease and disease-related deaths can lead to a new equilibrium population size below the carrying capacity and can even cause the population to become extinct.Research supported in part by Centers for Disease Control contract 200-87-0515     

An examination of the thresholds of enrichment: a resource-based growth model

A model of single-species growth in the chemostat on two non-reproducing, growth-limiting, noninhibitory, perfectly substitutable resources is considered. The medium in the growth vessel is enriched by increasing the input concentration of one of the resources. Analytical methods are used to determine the effects of enrichment on the asymptotic behaviour of the model for different dilution rates. It is shown that there exists a threshold value for the dilution rate which depends on the maximal growth rate of the species on each of the resources. Provided the dilution rate is below the threshold, enrichment is beneficial in the sense that the carrying capacity of the environment is increased, regardless of which resource is used to enrich the environment. When the dilution rate is increased beyond the threshold, it becomes important to consider which resource is used for enrichment. For one of the resources it is shown that, while moderate enrichment can be beneficial, sufficient enrichment leads to the extinction of the microbial population. For the other resource, enrichment leads from washout or initial condition dependent outcomes to survival, and is thus beneficial. There are important implications of these results to the management of natural aquatic ecosystems. For example, while enrichment may be beneficial to the microbial species during the summer months, it can lead to their decimation during spring run-off, when the natural dilution rate is higher.Research partially supported by an Ontario Graduate Scholarship. This author's contribution was motivated by results in her Ph.D. thesis at McMaster UniversityResearch supported by the National Sciences and Engineering Research Council of Canada.     

Permanence of single-species stage-structured models

In this paper, we consider population survival by using single-species stage-structured models. As a criterion of population survival, we employ the mathematical notation of permanence. Permanence of stage-structured models has already been studied by Cushing (1998). We generalize his result of permanence, and obtain a condition which guarantees that population survives. The condition is applicable to a wide class of stage-structured models. In particular, we apply our results to the Neubert-Caswell model, which is a typical stage-structured model, and obtain a condition for population survival of the model.The research is partially supported by the Ministry of Education, Science and Culture, Japan, under Grant in Aid for Scientific Research (A) 13304006.     

Predator-prey systems with group defence: The paradox of enrichment revisited

The main concern of this paper is with survival or extinction of predators in models of predator-prey systems exhibiting group defence of the prey. It is shown that if there is no mutual interference among predators, enrichment could result in their extinction. However, if there is mutual interference, the predator population survives (at least deterministically). Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. NSERC A 4823. Research partially supported by a Natural Sciences and Engineering Research Council of Canada postdoctoral fellowship.     

Simulation results with stepwise mutation model and their interpretations

Summary Monte Carlo simulations are performed to compare the predictions based on the two presently used theoretical models for studying genetic variations in natural populations, the infinite allele model and the stepwise mutation model. Distribution of heterozygosity is noticed to be similar under these models until the product of population size and mutation rate is large. It is seen that electromorphs with high population frequency usually contain older alleles (at the codon level) than an electromorph of low population frequency. The interpretations of these results in explaining the allelic variations at electrophoretic level is also discussed.Research supported by U.S. Public Health Service General Research Support Grant 5 SO 5 RR 07148 from the University of Texas Health Science Center, Graduate School of Biomedical Sciences, Houston, Texas     

A stochastic discrete generation birth,continuous death population growth model and its approximate solution

This paper describes a single species growth model with a stochastic population size dependent number of births occurring at discrete generation times and a continuous population size dependent death rate. An integral equation for a suitable transformation of the limiting population size density function is not in general soluble, but a Gram-Charlier representation procedure, previously used in storage theory, is successfully extended to cover this problem. Examples of logistic and Gompertz type growth are used to illustrate the procedure, and to compare with growth models in random environments. Comments on the biological consequences of these models are also given.Currently at Department of Mathematics, University of MarylandWork partially supported by the Danish Natural Science Research Council and Monash University     

Impacts of Incubation Delay on the Dynamics of an Eco-Epidemiological System—A Theoretical Study

Parasite and predator play significant role in trophic interaction, productivity and stability of an ecosystem. In this paper, we have studied a host-parasite-predator interaction that incorporates incubation delay. How the qualitative and quantitative behaviors of the system alter with the incubation delay have been discussed both from mathematical and biological point of views. It is observed that for a lower infection rate, the system is stable for all delays; but for a higher infection rate, there exists a threshold value of the delay above which the system is unstable and below which the system is stable leading to the persistence of all the species. Also, the instability arising from the incubation delay may be controlled if somehow the growth rate of predator population is increased. Numerical studies have also been performed to illustrate different analytical findings. Research is supported by UGC, India; F No. 32-173/2006(SR).     

The threshold of survival for system of two species in a polluted environment

The effects of toxicants on naturally stable two-species communities are studied. Persistence-extinction thresholds are given for populations in the toxicant stressed Lotka-Volterra model of two interacting species. The threshold results are expressed in terms of relationships involving the population intrinsic growth rates, dose-response parameters, and interaction rates.Research supported by the fund of Chinese Natural Science     

Analysis of the complicated dynamics of some harvesting models

In this paper we will study in a qualitative way discrete single species population models including harvesting. The class of models under consideration is quite general. In fact, we will study models with fixed parameter values. However, the obtained results do have implications for the models if one varies the parameters slightly. The models with so-called Allee-effect, i.e. the population will die out whenever the size of the population is below some threshold, are included in the class of models we studied.Research supported in part by the Netherlands organization for the advancement of pure research (Z.W.O.), a Fulbright grant, and a NSF grant. A part of this paper has been written while H.E.N. was visiting the Institute for Physical Science and Technology, University of Maryland, College Park, USA     

Some epidemiological models with nonlinear incidence

Epidemiological models with nonlinear incidence rates can have very different dynamic behaviors than those with the usual bilinear incidence rate. The first model considered here includes vital dynamics and a disease process where susceptibles become exposed, then infectious, then removed with temporary immunity and then susceptible again. When the equilibria and stability are investigated, it is found that multiple equilibria exist for some parameter values and periodic solutions can arise by Hopf bifurcation from the larger endemic equilibrium. Many results analogous to those in the first model are obtained for the second model which has a delay in the removed class but no exposed class.Research supported in part by Centers for Disease Control Contract 200-87-0515. Support services provided at University House Research Center at the University of IowaResearch supported in part by NSERC A-8965 and the University of Victoria President's Committee on Faculty Research and Travel     

Dynamic models of infectious diseases as regulators of population sizes

Five SIRS epidemiological models for populations of varying size are considered. The incidences of infection are given by mass action terms involving the number of infectives and either the number of susceptibles or the fraction of the population which is susceptible. When the population dynamics are immigration and deaths, thresholds are found which determine whether the disease dies out or approaches an endemic equilibrium. When the population dynamics are unbalanced births and deaths proportional to the population size, thresholds are found which determine whether the disease dies out or remains endemic and whether the population declines to zero, remains finite or grows exponentially. In these models the persistence of the disease and disease-related deaths can reduce the asymptotic population size or change the asymptotic behavior from exponential growth to exponential decay or approach to an equilibrium population size.Research supported by Centers for Disease Control contract 200-87-0515. Support services provided at the University of Iowa Center for Advanced Studies     

Allometric scaling and Bayesian priors for annual survival of birds and mammals

Allometric theory predicts that instantaneous mortality rates scale with body mass with a negative quarter power. Such a relationship would mean that the survival rate of one species is partly predictable from the survival rate of other species. We develop allometric regression models for annual adult survival of birds and mammals, using data collected from the literature. These models conform to the predictions of the allometric theory; the value of negative one-quarter for the scaling parameter is within the 95% credible interval, which is [-0.31, -0.10] for birds and [-0.35, -0.15] for mammals. The predictions are very well supported when evaluated using an independent set of data. The regression models can be used to provide objective and informative Bayesian priors for annual adult survival rates of birds and mammals or to act as a point of comparison in new studies.     

Estimation of growth and survival in size-structured cohort data: an application to larval striped bass (Morone saxatilis)

We present a method for estimating growth and mortality rates in size-structured population models. The methods are based on least-square fits to data using approximate models (using spline approximations) for the underlying partial differential equation population model. In a series of numerical tests, we compare our approach to an existing method (due to Hackney and Webb). As an example, we apply our techniques to experimental data from larval striped bass field studies.Research supported in part under grants at Brown University from the National Science Foundation: UINT-8521208, NSFDMS-8818530 (H.T.B., F.K. and CW.); from the Air Force Office of Scientific Research: AFOSR F49620-86-C-0111 (H.T.B., C.W.); and at University of California, Davis from the Alford P. Sloan Foundation (L.W.B.)     

Some stochastic features of bacterial constant growth apparatus

Stochastic models for bacterial constant growth apparatus such as the chemostat are posed and studied. Approximations are given for the mean and variance of the size of the bacterial population when the population is in steady state. Procedures for stimulating a chemostat are developed and the approximate moments are compared with simulated values. The distribution is derived for the waiting time until the occurrence of a population change-over to a faster growing strain. Research supported by National Institutes of Health Grant 5-R01-GM21214.     

On a conjecture concerning population growth in random environment

Discrete stochastic models are constructed and their limit diffusion processes are derived to shed light on a controversial conjecture regarding the effects of environmental variance on the asymptotic behavior of a population subject to logistic growth in random environment.Work supported in part by the National Group for Mathematical Information Sciences (GNIM) of the National Council for Research     

An analysis of neutral-alleles and variable-environment diffusion models

Three diffusion models are formulated for the evolution of a diploid population with K alleles at one locus with completely symmetric mutation and random genetic drift, a variable-environment, and all the above mechanisms. For the diallelic case, the transient behavior is studied by solving the corresponding diffusion equations by an asymptotic method valid for short time intervals. The transient behavior of the three models is compared for the case when their stationary distributions are identical. The expected amount of heterozygosity is computed using the asymptotic solution and is compared to an exact result. The asymptotic results are extended to the general case with K alleles at the locus for the symmetric mutation and variable-environment models.Research supported by the National Science Foundation under Grant MCS 79-01718     

温度对番茄潜叶蛾生长发育和繁殖的影响

【目的】番茄潜叶蛾Tuta absoluta是一种对番茄具有毁灭性危害的世界性入侵害虫,本研究旨在探索温度对入侵番茄潜叶蛾种群生长发育和繁殖的影响,为预测番茄潜叶蛾的分布区域、田间发生动态提供基础。【方法】在室内测定了番茄叶片上番茄潜叶蛾在15, 20, 25, 30和35℃ 5个恒定温度条件下各虫态的发育历期和存活率、繁殖力和种群增长参数,并应用不同模型分析发育速率、内禀增长率和净生殖率与温度的关系,估计发育起点温度、发育极限温度、发育最适温度、有效积温和年发生代数。【结果】在恒温15~30℃范围内,番茄潜叶蛾各虫态的发育历期随温度升高而逐渐缩短。25℃下幼虫期存活率、成虫前期存活率、雌虫总产卵量、净生殖率、内禀增长率和周限增长率均最大。在35℃下,卵的存活率骤降至11%,孵化的幼虫无法正常发育。卵期、幼虫期、蛹期、成虫前期、全世代的有效积温分别为104.17, 232.59, 129.87, 434.78和526.32日·度,该虫在新疆伊宁县和察布查尔锡伯自治县的理论发生代数为4~5代。基于发育速率与温度关系的模型与基于种群增长参数与温度关系的模型所计算的积温需求参数不同,基于内禀增长率求得的番茄潜叶蛾的发育起点温度、发育极限高温和发育最适温度分别为12.46, 30.40和27.36℃。【结论】入侵我国新疆地区的番茄潜叶蛾适温范围广泛,在我国大部分地区具有极高的扩散风险。     

Estimating Allee Dynamics before They Can Be Observed: Polar Bears as a Case Study

Allee effects are an important component in the population dynamics of numerous species. Accounting for these Allee effects in population viability analyses generally requires estimates of low-density population growth rates, but such data are unavailable for most species and particularly difficult to obtain for large mammals. Here, we present a mechanistic modeling framework that allows estimating the expected low-density growth rates under a mate-finding Allee effect before the Allee effect occurs or can be observed. The approach relies on representing the mechanisms causing the Allee effect in a process-based model, which can be parameterized and validated from data on the mechanisms rather than data on population growth. We illustrate the approach using polar bears (Ursus maritimus), and estimate their expected low-density growth by linking a mating dynamics model to a matrix projection model. The Allee threshold, defined as the population density below which growth becomes negative, is shown to depend on age-structure, sex ratio, and the life history parameters determining reproduction and survival. The Allee threshold is thus both density- and frequency-dependent. Sensitivity analyses of the Allee threshold show that different combinations of the parameters determining reproduction and survival can lead to differing Allee thresholds, even if these differing combinations imply the same stable-stage population growth rate. The approach further shows how mate-limitation can induce long transient dynamics, even in populations that eventually grow to carrying capacity. Applying the models to the overharvested low-density polar bear population of Viscount Melville Sound, Canada, shows that a mate-finding Allee effect is a plausible mechanism for slow recovery of this population. Our approach is generalizable to any mating system and life cycle, and could aid proactive management and conservation strategies, for example, by providing a priori estimates of minimum conservation targets for rare species or minimum eradication targets for pests and invasive species.     

Drought survival is a threshold function of habitat size and population density in a fish metapopulation

Because smaller habitats dry more frequently and severely during droughts, habitat size is likely a key driver of survival in populations during climate change and associated increased extreme drought frequency. Here, we show that survival in populations during droughts is a threshold function of habitat size driven by an interaction with population density in metapopulations of the forest pool dwelling fish, Neochanna apoda. A mark–recapture study involving 830 N. apoda individuals during a one‐in‐seventy‐year extreme drought revealed that survival during droughts was high for populations occupying pools deeper than 139 mm, but declined steeply in shallower pools. This threshold was caused by an interaction between increasing population density and drought magnitude associated with decreasing habitat size, which acted synergistically to increase physiological stress and mortality. This confirmed two long‐held hypotheses, firstly concerning the interactive role of population density and physiological stress, herein driven by habitat size, and secondly, the occurrence of drought survival thresholds. Our results demonstrate how survival in populations during droughts will depend strongly on habitat size and highlight that minimum habitat size thresholds will likely be required to maximize survival as the frequency and intensity of droughts are projected to increase as a result of global climate change.