Dynamical behavior of differential equation models of frequency and density dependent populations

This paper studies the dynamics of a mathematical model of a continuously reproducing diploid population with two alleles at one locus. The dependent variables are allele frequency and population density. If the genotype fitnesses are frequency and density dependent, the stability of equilibria is related to the geometry of the zero allele fitness curves. The asymptotic behavior of solutions where fitness is only density dependent is contrasted to the asymptotic behavior where fitness is frequency and density dependent. A parameterized family of fitness functions giving a Hopf bifurcation and limit cycles is investigated analytically and numerically.     

Effect of predator density dependent dispersal of prey on stability of a predator-prey system

This work presents a predator-prey Lotka-Volterra model in a two patch environment. The model is a set of four ordinary differential equations that govern the prey and predator population densities on each patch. Predators disperse with constant migration rates, while prey dispersal is predator density-dependent. When the predator density is large, the dispersal of prey is more likely to occur. We assume that prey and predator dispersal is faster than the local predator-prey interaction on each patch. Thus, we take advantage of two time scales in order to reduce the complete model to a system of two equations governing the total prey and predator densities. The stability analysis of the aggregated model shows that a unique strictly positive equilibrium exists. This equilibrium may be stable or unstable. A Hopf bifurcation may occur, leading the equilibrium to be a centre. If the two patches are similar, the predator density dependent dispersal of prey has a stabilizing effect on the predator-prey system.     

The computer-aided bifurcation analysis of predator-prey models

Mathematical analysis of dynamical systems can often benefit from accompanying numerical computations. This is particularly true if one has software (e.g. AUTO [6, 7]) capable of providing an automatic bifurcation analysis of such systems. Computer programs of this type now exist. We describe the application of such software to a predator-prey model. Phenomena that arise in this analysis include stationary bifurcations, limit points, Hopf bifurcations and secondary periodic bifurcations. A two-parameter numerical analysis leads quite naturally to the detection of higher order singularities.Supported in part by NSERC Canada (#4274) and FCAC Québec (#EQ1438)     

Bifurcation structure of a chemostat model for an age-structured predator and its prey

We model a chemostat containing an age-structured predator and its prey using a linear function for the uptake of substrate by the prey and two different functional responses (linear and Monod) for the consumption of prey by the predator. Limit cycles (LCs) caused by the predator's age structure arise at Hopf bifurcations at low values of the chemostat dilution rate for both model cases. In addition, LCs caused by the predator–prey interaction arise for the case with the Monod functional response. At low dilution rates in the Monod case, the age structure causes cycling at lower values of the inflowing resource concentration and conversely prevents cycling at higher values of the inflowing resource concentration. The results shed light on a similar model by Fussmann et al. [G. Fussmann, S. Ellner, K. Shertzer, and N. Hairston, Crossing the Hopf bifurcation in a live predator–prey system, Science 290 (2000), pp. 1358–1360.], which correctly predicted conditions for the onset of cycling in a chemostat containing an age-structured rotifer population feeding on algal prey.     

Effects of prey density,prey size and predator size on rates of feeding by an intertidal predatory gastropod Morula marginalba Blainville (Muricidae), on several species of prey

As a prerequisite for models of foraging behaviour of the whelk, Morula marginalba Blainville (Muricidae), the effects of variation in density of prey on the rate of feeding of the predator were examined in field conditions for three coexisting species of prey. Densities of prey used were those at which the prey, two limpets and a barnacle, occurred naturally in the rocky intertidal habitat.Large limpets, Cellana tramoserica (Sowerby) can resist attacks by predatory gastropods by raising the mantle over the outside of the shell. These experiments showed that no C. tramoserica were killed by Morula marginalba even at very great densities and with no alternative prey present. For the small limpet Patelloida latistrigata (Angas), one of the whelk's most highly preferred prey, juveniles were eaten 1.4 times as fast as adults. Fitting the random predator equation gave greater attack coefficients and shorter handling times for juvenile than adult limpets.Sizes of both predator and prey affected rates of eating barnacles, Tesseropora rosea (Krauss), but not in a simple way. Whelks of 15-mm aperture length ate adult barnacles 4.2 times faster than did 12-mm whelks, but there was no significant difference in the rates at which the two sizes of snail ate juvenile barnacles.Rates of feeding on T. rosea and Patelloida latistrigata increased significantly with prey density. These results form a basis for including the density of prey in models of spatial dispersion of the predatory gastropod Morula marginalba.     

Almost periodic solution of non-autonomous Lotka-Volterra predator-prey dispersal system with delays

This paper studies a non-autonomous Lotka-Volterra almost periodic predator-prey dispersal system with discrete and continuous time delays which consists of n-patches, the prey species can disperse among n-patches, but the predator species is confined to one patch and cannot disperse. By using comparison theorem and delay differential equation basic theory, we prove the system is uniformly persistent under some appropriate conditions. Further, by constructing suitable Lyapunov functional, we show that the system is globally asymptotically stable under some appropriate conditions. By using almost periodic functional hull theory, we show that the almost periodic system has a unique globally asymptotical stable strictly positive almost periodic solution. The conditions for the permanence, global stability of system and the existence, uniqueness of positive almost periodic solution depend on delays, so, time delays are "profitless". Finally, conclusions and two particular cases are given. These results are basically an extension of the known results for non-autonomous Lotka-Volterra systems.     

Complex dynamics in biological systems arising from multiple limit cycle bifurcation

In this paper, we study complex dynamical behaviour in biological systems due to multiple limit cycles bifurcation. We use simple epidemic and predator–prey models to show exact routes to new types of bistability, that is, bistability between equilibrium and periodic oscillation, and bistability between two oscillations, which may more realistically describe the real situations. Bifurcation theory and normal form theory are applied to investigate the multiple limit cycles bifurcating from Hopf critical point.     

Self-optimization, community stability, and fluctuations in two individual-based models of biological coevolution

We compare and contrast the long-time dynamical properties of two individual-based models of biological coevolution. Selection occurs via multispecies, stochastic population dynamics with reproduction probabilities that depend nonlinearly on the population densities of all species resident in the community. New species are introduced through mutation. Both models are amenable to exact linear stability analysis, and we compare the analytic results with large-scale kinetic Monte Carlo simulations, obtaining the population size as a function of an average interspecies interaction strength. Over time, the models self-optimize through mutation and selection to approximately maximize a community potential function, subject only to constraints internal to the particular model. If the interspecies interactions are randomly distributed on an interval including positive values, the system evolves toward self-sustaining, mutualistic communities. In contrast, for the predator–prey case the matrix of interactions is antisymmetric, and a nonzero population size must be sustained by an external resource. Time series of the diversity and population size for both models show approximate 1/f noise and power-law distributions for the lifetimes of communities and species. For the mutualistic model, these two lifetime distributions have the same exponent, while their exponents are different for the predator–prey model. The difference is probably due to greater resilience toward mass extinctions in the food-web like communities produced by the predator–prey model.      

Stability of periodic solutions for a model of genetic repression with delays

A technique is discussed for locating the Hopf bifurcation of an n-dimensional system of delay differential equations which arises from a model for control of protein biosynthesis. Certain parameter values are shown to allow a Hopf bifurcation to periodic orbits. At the Hopf bifurcation the periodic orbits are shown to be stable either analytically or numerically depending on the parameter values.On leave from North Carolina State University.Supported in part by N.S.F. Grant # MCS 81-02828     

Adjacent trophic-level effects on spatial density dependence in a herbivore—predator—parasitoid system

Abstract. 1. Spatial density dependence of enemy/victim relationships (predator/prey and host/parasitoid) were examined for the gall-making herbivore Eurosta solidaginis (Diptera: Tephritidae), its predator Mordellistena unicolor (Coleoptera: Mordellidae), and a parasitoid of Mordellistena, Schizoprymnus sp. (Hymenoptera: Braconidae), both before and after experimental perturbation of gall density.
2. Mordellistena predation did not depend on Eurosta density, nor did Schizoprymnus parasitism of Mordellistena depend on Mordellistena density.
3. Schizoprymnus parasitism of Mordellistena depended strongly on the density of Eurosta , the prey of its host.
4. The lack of density dependence in enemy/victim relationships may be explained by effects from adjacent trophic-levels. Ovipositing Schizoprymnus may search for high densities, not of its host Mordellistena , but of its host's prey ( Eurosta galls), and ovipositing Mordellistena may avoid patches of high gall density where the risk of being parasitized is greater.     

Population dynamics of three songbird species in a nestbox population in Central Europe show effects of density,climate and competitive interactions

Unravelling the contributions of density‐dependent and density‐independent factors in determining species population dynamics is a challenge, especially if the two factors interact. One approach is to apply stochastic population models to long‐term data, yet few studies have included interactions between density‐dependent and density‐independent factors, or explored more than one type of stochastic population model. However, both are important because model choice critically affects inference on population dynamics and stability. Here, we used a multiple models approach and applied log‐linear and non‐linear stochastic population models to time series (spanning 29 years) on the population growth rates of Blue Tits Cyanistes caeruleus, Great Tits Parus major and Pied Flycatchers Ficedula hypoleuca breeding in two nestbox populations in southern Germany. We focused on the roles of climate conditions and intra‐ and interspecific competition in determining population growth rates. Density dependence was evident in all populations. For Blue Tits in one population and for Great Tits in both populations, addition of a density‐independent factor improved model fit. At one location, Blue Tit population growth rate increased following warmer winters, whereas Great Tit population growth rates decreased following warmer springs. Importantly, Great Tit population growth rate also decreased following years of high Blue Tit abundance, but not vice versa. This finding is consistent with asymmetric interspecific competition and implies that competition could carry over to influence population dynamics. At the other location, Great Tit population growth rate decreased following years of high Pied Flycatcher abundance but only when Great Tit population numbers were low, illustrating that the roles of density‐dependent and density‐independent factors are not necessarily mutually exclusive. The dynamics of this Great Tit population, in contrast to the other populations, were unstable and chaotic, raising the question of whether interactions between density‐dependent and density‐independent factors play a role in determining the (in) stability of the dynamics of species populations.     

Predators reverse the direction of density dependence for juvenile salmon mortality

The effect of predators on prey populations depends on how predator-caused mortality changes with prey population density. Predators can enforce density-dependent prey mortality and contribute to population stability, but only if they have a positive numerical or behavioral response to increased prey density. Otherwise, predator saturation can result in inversely density-dependent mortality, destabilizing prey populations and increasing extinction risk. Juvenile salmon and trout provide some of the clearest empirical examples of density-dependent mortality in animal populations. However, although juvenile salmon are very vulnerable to predators, the demographic effects of predators on juvenile salmon are unknown. We tested the interactive effects of predators and population density on the mortality of juvenile Atlantic salmon (Salmo salar) using controlled releases of salmon in natural streams. We introduced newly hatched juvenile salmon at three population density treatments in six study streams, half of which contained slimy sculpin (Cottus cognatus), a common generalist predator (18 release sites in total, repeated over two summers). Sculpin reversed the direction of density dependence for juvenile salmon mortality. Salmon mortality was density dependent in streams with no sculpin, but inversely density dependent in streams where sculpin were abundant. Such predator-mediated inverse density dependence is especially problematic for prey populations suppressed by other factors, thereby presenting a fundamental challenge to persistence of rare populations and restoration of extirpated populations.     

非自治阶段结构合作系统的持久性与周期解

本文研究一类非自治阶段结构的合作系统,得到系统的最终有界性,对应周期系统正周期解的存在性,唯一性以及全局渐近稳定性的充分条件。     

Prey selection by a spider wasp,Batozonellus lacerticida (Hymenoptera: Pompilidae): Effects of seasonal variation in prey species,size and density

Prey selection by a spider wasp,Batozonellus lacerticida, was investigated at the riverside of the Toyohira River in Misumai, Sapporo, Hokkaido over 7 years from 1981 to 1987. Seventeen species of araneid spiders were found in the study area and six of them were hunted by the wasps. Most (97.3%) of the 223 prey records obtained were of three species,Araneus macacus, A. marmoreus andA. pinguis. Nesting activity of the wasps continued until mid-August in 1981, 1982 and 1986, but stopped in July in 1984, 1985 and 1987. The length of nesting period was related to whether the wasps successfully switched prey species fromA. macacus toA. marmoreus andA. pinguis andA. pinguis or not Switching seemed to occur at the phase when density ofA. macacus remained high, depending on the density of largerA. marmorus andA. pinguis. This switching had large effects on species composition and size distribution of actual prey. Chesson's index α calculated at every 10 day period revealed that the wasps preferredA. macacus to the two other species and large prey to medium and small ones. The effect of prey density on preference depended on whetherA. macacus was present or absent. In the presence of it there were only slight differences in preference among various conditions of prey density. However, whenA. macacus was absent, densities of bothA. marmoreus andA. pinguis had some complementary effects on preference. The importance of prey size selection by pompilid wasps, and implications of density effects on preference are discussed.     

具有扩散的非自治两种群Lotka-Volterra模型的概周期问题

研究非自治两种群竞争系统,其中一种群可以在两个斑块之间扩散。而另一种群在一个斑块中,不能扩散。本文结合运用Liapunov函数,得到该系统唯一存在全局渐近稳定的正概周期解的条件．     

No evidence of nonlinear effects of predator density,refuge availability,or body size of prey on prey mortality rates

Predator density, refuge availability, and body size of prey can all affect the mortality rate of prey. We assume that more predators will lead to an increase in prey mortality rate, but behavioral interactions between predators and prey, and availability of refuge, may lead to nonlinear effects of increased number of predators on prey mortality rates. We tested for nonlinear effects in prey mortality rates in a mesocosm experiment with different size classes of western mosquitofish (Gambusia affinis) as the prey, different numbers of green sunfish (Lepomis cyanellus) as the predators, and different levels of refuge. Predator number and size class of prey, but not refuge availability, had significant effects on the mortality rate of prey. Change in mortality rate of prey was linear and equal across the range of predator numbers. Each new predator increased the mortality rate by about 10% overall, and mortality rates were higher for smaller size classes. Predator–prey interactions at the individual level may not scale up to create nonlinearity in prey mortality rates with increasing predator density at the population level.     

Analysis of a time-delayed mathematical model for tumour growth with an almost periodic supply of external nutrients

In this paper, the existence, uniqueness and exponential stability of almost periodic solutions for a mathematical model of tumour growth are studied. The establishment of the model is based on the reaction–diffusion dynamics and mass conservation law and is considered with a delay in the cell proliferation process. Using a fixed-point theorem in cones, the existence and uniqueness of almost periodic solutions for different parameter values of the model is proved. Moreover, by the Gronwall inequality, sufficient conditions are established for the exponential stability of the unique almost periodic solution. Results are illustrated by computer simulations.     

Effect of prey density on sex ratio of two predacious mites, Phytoseiulus persimilis and Amblyseius womersleyi (Acari: Phytoseiidae)

The sex ratios of two phytoseiid mites, Phytoseiulus persimilis and Amblyseius womersleyi, were observed under various prey conditions. Upon consumption of abundant prey, both phytoseiids produced progeny in a female-biased sex ratio (approximately 0.8 females). When few prey were consumed, the sex ratio was lowered to 0.5 (the unbiased sex ratio). Under the conditions in which the unbiased sex ratio was observed, male and female progeny appeared in an alternating sequence. To determine the change in the sex ratio and the sequence of progeny, the size of eggs deposited by females under various prey conditions was first compared. Survivorship and developmental rate of progeny (immatures) hatched under ample and poor prey conditions were also examined. The eggs deposited under poor prey conditions were smaller than those deposited under ample prey conditions. This is an indication that the phytoseiid females did not invest extra energy into the eggs to secure survival of their progeny under poor prey conditions. The male and female progeny from the small eggs developed slowly, probably due to the small egg size. However, hatchability of the small eggs and survival of the immatures were the same as those of the normal eggs when the immatures were reared under ample prey conditions. The immature survivorship was little affected by the prey consumption rate of their mothers when the immatures were reared under poor prey conditions. We concluded that the sex ratio of phytoseiid mites is not determined by the characteristics of the progeny, but by the nutritious condition of the females.Exp Appl Acarol 22: 709723 © 1998 Kluwer Academic Publishers     

Negative density dependence can offset the effect of species competitive asymmetry: a niche-based mechanism for neutral-like patterns

The debate on the role of species differences in shaping biodiversity patterns, with its two extremes of pure niche theory and neutral theory, is still ongoing. It has been demonstrated that a slight difference in competitive ability of species severely affects the predictions of the neutral model. At the same time, neutral patterns seem to be ubiquitous. Here, we model both negative density dependence (NDD) and competitive asymmetry (CA) simultaneously. Our simulation results show that an appropriate intensity of NDD can offset the negative effect of CA (modeled as fecundity difference) on species coexistence and produce a neutral-like species abundance distribution. Therefore, our model provides a plausible mechanistic explanation of neutral-like patterns, but contrary to the neutral model, a species' relative abundance is positively related to its competitive ability in our model.     

Interaction of pH,density, and priority effects on the survivorship and growth of two species of hylid tadpoles

Summary We examined the interactions of an abiotic factor (pH) and a biotic factor (density) on the survival and growth of two species of anuran larvae (Hyla gratiosa and Hyla femoralis) in outdoor tanks. Three levels of pH (4.3, 4.6, or 6.0) and three levels of density (0, 30 or 60 embryos) were arranged in a blocked design and replicated three times for Hyla gratiosa. At the end of this experiment the effects of pH (4.3, 4.6, or 6.0), density of H. femoralis (30 or 60), and prior use by H. gratiosa (at 0, 30, or 60 larvae per tank) on the survival and growth of H. femoralis, were examined. Higher density increased larval period and decreased size at metamorphosis of H. gratiosa. Lower pH decreased survival rate and also decreased size at metamorphosis. Body sodium concentrations were lowest at the low pH values. Lower pH increased the susceptibility of H. gratiosa tadpoles to the adverse effects of higher densities. For H. femoralis higher density decreased survival, increased larval period and decreased size at metamorphosis. Hyla femoralis also had lower survivorship at low pH and exhibited decreased size at metamorphosis. However, unlike the results with Hyla gratiosa, there were no interactive effects between pH and density for any of the life-history traits studied. The effect of previous colonization by H. gratiosa on H. femoralis survival was facilitative. Body sodium concentrations of H. femoralis were lowest at the highest pH value. Metamorphs of the same size had much lower levels of sodium in H. femoralis than H. gratiosa. In general, H. femoralis was less affected by pH variation than H. gratiosa. These results demonstrate that abiotic factors can interact strongly with biotic effects such as density and they suggest that interspecific interactions can be strongly modulated by the background abiotic environment.