Invasion of an intermediate predator: Fish population dynamics in a mathematical model of a trophic chain (As applied to Syamozero lake)

A mathematical model is presented for the dynamics of a spatially heterogeneous predator-prey population system; a prototype is the Syamozero lake fish community. We show that the invasion of an intermediate predator can evoke chaotic oscillations in the population densities. We also show that different dynamic regimes (stationary, nonchaotic oscillatory, and chaotic) can coexist. The “choice” of a particular regime depends on the initial invader density. Analysis of the model solutions shows that invasion of an alien species is successful only in the absence of competition between the juvenile invaders and the native species.     

一类随机时滞捕食者-食饵模型的动力学行为（英文）

研究了一类含时滞的Harrison型捕食者-食饵模型在随机扰动环境下的动力学行为.对于非时滞和时滞模型分别给出了局部和全局稳定性条件.通过白噪声分别对食饵人口增长率的和捕食者人口死亡率进行随机扰动,构建相应的随机时滞微分方程模型讨论环境噪声对其作用的动力学行为.在一定条件下,随机时滞模型存在随机最终有界的唯一全局正解且解的二阶均值是有界的.最后通过数值模拟对给出的分析结果进行了验证.     

Individual base model of predator-prey system shows predator dominant dynamics

Individual base model of predator-prey system is constructed. Both predator and prey species have age structure and cohorts of early reproductive age have competitive advantage. The model has linear functional response in predation behavior and includes the effect of interference among predators and delay of population growth from resource intake, not by functional response but by calculation procedure. Each foraging action is calculated successively and surplus or scarce of acquired resources is interpreted into population size through individual birth and death. This model shows that biomass of prey killed by predator is dependent on demand of predator and that heterogeneity in predator population is essential in persistency and stability of predator-prey system. Heterogeneity of predator makes predator individuals of less competing ability die rapidly. Rapid death of weak individuals causes rapid decrease of total demand of predator and that makes enough room for survived predators. Therefore, the biomass of killed prey is dependent on predator's demand. As young or infant population of predator are the more vulnerable to shortage of prey, and when many of them cannot survive to reproductive age, they can stabilize the system by wasting excessive prey with only temporal numerical increase of predator population.     

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.     

Bifurcating periodic solutions for a class of age-structured predator-prey systems

A system of mixed integrodifferential and partial differential equations for an agestructured predator-prey system is studied here. The predator eats all ages of prey, but more of the very young and very old than of the intermediate ages. The existence of periodic solutions corresponding to stable coexistence is proved for a suitable range of parameters by bifurcation theory.     

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.     

Effect of delay in a Lotka-Volterra type predator-prey model with a transmissible disease in the predator species

We consider a system of delay differential equations modeling the predator-prey ecoepidemic dynamics with a transmissible disease in the predator population. The time lag in the delay terms represents the predator gestation period. We analyze essential mathematical features of the proposed model such as local and global stability and in addition study the bifurcations arising in some selected situations. Threshold values for a few parameters determining the feasibility and stability conditions of some equilibria are discovered and similarly a threshold is identified for the disease to die out. The parameter thresholds under which the system admits a Hopf bifurcation are investigated both in the presence of zero and non-zero time lag. Numerical simulations support our theoretical analysis.     

Sigmoid functional responses and population stability

Sigmoid functional responses are known to stabilize the differential Lotka-Volterra predator-prey model. However, we have found that they have no such effect in a comparable discrete generation model. The difficulty in stabilizing this model results from the one-generation time delay between changes in predator population density and the level of prey mortality. By contrast, sigmoid functional responses can stabilize the system if the predator population remains relatively constant, as is more likely of generalist predators.     

Mothers' Egg-pooling and Aposematic Offspring's Gregariousness: Cui bono?

Recently Sillén-Tullberg & Leimar (1988) modelled a general explanation for the evolution of gregariousness in prey organisms that live exposed, have no means of escape when discovered by a predator, and are small in relation to a potential predator (who thus can sample many prey individuals in one encounter). The model predicts that gregarious prey organisms of that type ought to be distasteful, and that the evolution of gregariousness will be favoured by aposematic coloration facilitating avoidance learning in a predator. Obviously, any protective power of grouping depends on group size. According to the Sillén-Tullberg & Leimar model, (1) “members of small groups may have a higher rate of death from predation than solitary individuals, but above a certain minimum group size, group members do better than solitary individuals; … as group size increases above the minimum value, group members suffer fewer and fewer deaths from predation”. They benefit from the “decreased risk of predator attack on any particular individual”, called dilution effect. (2) “The more prey specimens that the predator needs to sample during avoidance learning, the larger an aggregation needs to be in order for gregariousness to be advantageous”. It is further explained that (3) selection resulting from predation favours increase in group size until it “acts like a predator-satiation mechanism”.     

Effects of a disease affecting a predator on the dynamics of a predator-prey system

We study the effects of a disease affecting a predator on the dynamics of a predator-prey system. We couple an SIRS model applied to the predator population, to a Lotka-Volterra model. The SIRS model describes the spread of the disease in a predator population subdivided into susceptible, infected and removed individuals. The Lotka-Volterra model describes the predator-prey interactions. We consider two time scales, a fast one for the disease and a comparatively slow one for predator-prey interactions and for predator mortality. We use the classical “aggregation method” in order to obtain a reduced equivalent model. We show that there are two possible asymptotic behaviors: either the predator population dies out and the prey tends to its carrying capacity, or the predator and prey coexist. In this latter case, the predator population tends either to a “disease-free” or to a “disease-endemic” state. Moreover, the total predator density in the disease-endemic state is greater than the predator density in the “disease-free” equilibrium (DFE).     

Explicit model for searching behavior of predator

The authors present an approach for explicit modeling of spatio-temporal dynamics of predator-prey community. This approach is based on a reaction-diffusion-adjection PD (prey dependent) system. Local kinetics of population is determined by logistic reproduction function of prey, constant natural mortality of predator and Holling type 2 trophic function. Searching behavior of predator is described by the advective term in predator balance equation assuming the predator acceleration to be proportional to the prey density gradient. The model was studied with zero-flux boundary conditions. The influence of predator searching activity on the community dynamics, in particular, on the emergence of spatial heterogeneity, has been investigated by linear analysis and numerical simulations. It has been shown how searching activity may effect the persistence of species, stabilizing predator-prey interactions at very low level of pest density. It has been demonstrated that obtaining of such dynamic regimes does not require the use of complex trophic functions.     

The cultivation and predator-prey relationships Chænæ vorax and Cyclidium sp.

Chaenea vorax, a predatory marine gymnostome, was established in laboratory culture using Cyclidium sp. as a food source. The respective cultures were used to study the predator-prey relationships of these organisms. In addition, the feeding specifics of Chaenea vorax were studied and a possible sexual phenomenon, herein called “coupling”, was observed. Whenever Chaenea vorax is introduced into a laboratory culture of Cyclidium sp. the predator brings about a decimation of the prey population. Other food sources were offered to the predator, but only those organisms having a long caudal cilium were used and/or grasped. The phenomenon of “coupling” is proposed to be conjugation, but the nuclear events have not been established.     

Identifying predator-prey processes from time-series

The functional response is a key element in predator-prey models as well as in food chains and food webs. Classical models consider it as a function of prey abundance only. However, many mechanisms can lead to predator dependence, and there is increasing evidence for the importance of this dependence. Identification of the mathematical form of the functional response from real data is therefore a challenging task. In this paper we apply model-fitting to test if typical ecological predator-prey time series data, which contain both observation error and process error, can give some information about the form of the functional response. Working with artificial data (for which the functional response is known) we will show that with moderate noise levels, identification of the model that generated the data is possible. However, the noise levels prevailing in real ecological time-series can give rise to wrong identifications. We will also discuss the quality of parameter estimation by fitting differential equations to such time-series.     

捕食者-猎物关系的理论和应用研究

捕食者和猎物相互关系的研究,长期以来一直是动物生态学研究的中心课题之一。这种研究可区分为3种类型:第一是理论研究,即组建各种数学模型以便模拟捕食者和猎物间的相互关系;第二是实验种群研究,通常是在实验室内选用原生动物和节肢动物实验种群对捕食一猎物间的动态关系进行观察,并将观察结果与理论模型进行比较;第三是在田间对自然种群中的捕食者-猎物关系进行研究,并利用从理论研究和实验种群研究中所总结出来的各种基本原理对观察资料进行分析。在应用生态学领域中,常常靠引进新的更加有效的天敌来控制各种害虫。这些实际工作有些已获得成功,也有不少未取得预期效果,不管成功与否,它们都可被看作是对捕食者-猎物关系所进行的田间实验.     

Stability analysis of a predator-prey system with mutual interference and density-dependent death rates

A two species predator-prey model is proposed incorporating the notions of mutual interference among predators as well as a density-dependent predator death rate. The latter leads to a curved predator isocline. Conditions for an interior equilibrium are given, and the stability of this equilibrium is analyzed. Certain critical cases, some of which cannot occur in the usual model are also discussed.     

ADAPTIVE RESPONSES OF PREDATORS TO PREY AND PREY TO PREDATORS: THE FAILURE OF THE ARMS-RACE ANALOGY

This paper analyzes a number of relatively general models of predator-prey adaptation and coadaptation. The motivation behind this work is, in part, to evaluate the “race analogy” that has been applied in analyzing predator-prey coevolution. The models are based on the assumption that increased investment in predation-related adaptations must be paid for by decreased adaptation to some other factor. Increased investment in predation-related adaptations by the prey lowers the predator's functional response, and increased investment by the predator increases the functional response. The models are used to determine how each species should respond to an increase in the predation-related investment of the other species. Several broad classes of population-dynamics models and several alternatives for the cost of predation-related adaptation are investigated. The results do not support the general applicability of the race analogy. In the type of model analyzed in greatest detail here, predator and prey adaptations combine multiplicatively in determining the predator's capture-rate constant. In such models, prey usually increase investment in predator avoidance or escape when predators increase their investment in capture. However, predators often do not change or decrease their investment in response to an increase in the prey's investment. The direction of the predator's response depends on the particular parameter that pays the cost of increased predation investment, the shape of the cost-benefit functions, and the assumptions about the population dynamics of the predator-prey system. Similar models are used to determine whether increased investment by one species should increase the rate of incorporation of mutations that improve the predation-related adaptations of the other species. The arms-race analogy also fails for this case. The results cast doubt on the usefulness of Dawkins and Krebs (1979) “life-dinner” principle.     

一类具分段常数变量的捕食-食饵系统的Neimark-Sacker分支

讨论了具时滞与分段常数变量的捕食-食饵生态模型的稳定性及Neimark-Sacker分支;通过计算得到连续模型对应的差分模型,基于特征值理论和Schur-Cohn判据得到正平衡态局部渐进稳定的充分条件;以食饵的内禀增长率为分支参数,运用分支理论和中心流形定理分析了Neimark-Sacker分支的存在性与稳定性条件;通过举例和数值模拟验证了理论的正确性。     

A detailed study of the Beddington-DeAngelis predator-prey model

The present investigation accounts for the influence of intra-specific competition among predators in the original Beddington-DeAngelis predator-prey model. We offer a detailed mathematical analysis of the model to describe some of the significant results that may be expected to arise from the interplay of deterministic and stochastic biological phenomena and processes. In particular, stability (local and global) and bifurcation (Saddle-node, Transcritical, Hopf-Andronov, Bogdanov-Takens) analysis of this model are conducted. Corresponding results from previous well known predator-prey models are compared with the current findings. Nevertheless, we also allow this model in stochastic environment with the influences of both, uncorrelated “white” noise and correlated “coloured” noise. This showing that competition among the predator population is beneficial for a number of predator-prey models by keeping them stable around its positive interior equilibrium (i.e. when both populations co-exist), under environmental stochasticity. Comparisons of these findings with the results of some earlier related investigations allow the general conclusion that predator intra-species competition benefits the predator-prey system under both deterministic and stochastic environments. Finally, an extended discussion of the ecological implications of the analytical and numerical results concludes the paper.     

食饵庇护所对斑块环境下Leslie-Gower捕食系统的影响

建立了一个显式含有空间庇护所的两斑块Leslie-Gower捕食者-食饵系统。假设只有食饵种群在斑块间以常数迁移率迁移,且在每个斑块上食饵间的迁移比局部捕食者-食饵相互作用发生的时间尺度要快。利用两个时间尺度,可以构建用来描述所有斑块总的食饵和捕食者密度的综合系统。数学分析表明,在一定条件下,存在唯一的正平衡点,并且此平衡点全局稳定。进一步,捕食者的数量随着食饵庇护所数量增加而降低;在一定条件下,食饵的数量随着食饵庇护所数量增加先增加后降低,在足够强的庇护所强度下,两物种出现灭绝。对比以往研究,利用显式含有和隐含空间庇护所的数学模型所得结论不一致,这意味着在研究庇护所对捕食系统种群动态影响时,空间结构可能起着重要作用。