Influence of stochastic perturbation on prey-predator systems

We analyse the influence of various stochastic perturbations on prey-predator systems. The prey-predator model is described by stochastic versions of a deterministic Lotka-Volterra system. We study long-time behaviour of both trajectories and distributions of the solutions. We indicate the differences between the deterministic and stochastic models.     

Nonlinear phenomena and chaos in a Monte Carlo simulated microbial ecosystem

Oscillations and chaos can be modelled and observed in a realistic simulation model of interacting prey-predator populations based on Monte Carlo simulation methods. These nonlinear phenomena are linked with some biological and physical bifurcation parameters and mathematical tools from dynamical systems theory may be used in order to characterize this behaviour. Chaotic dynamics are therefore, in our simulation, more the rule than the exception, and are related to delays associated with spatial degrees of freedom.     

具有庇护所的三种群捕食者-食饵模型

讨论了一类具有庇护所的自治三种群捕食者一食饵模型,运用Liapunov函数方法,得到了该模型持久性的充分条件．对于该模型的周期系统,在一定的条件下,将产生唯一一个全局渐近稳定的周期正解．对更具普遍意义的概周期现象,也得出了概周期正解唯一存在且全局渐近稳定性的充分条件．     

非齐次的食饵——捕食模型

为了建立被捕食与捕食模型,我们在ｚ＾ｄ上考虑两种群的相互作用,还考虑了在被捕食者数目很大时,捕食者变化地趋于一个常数的情况,最后,给出了关于两种群共存的与非共存问题的研究。     

广义Lotke-Volterra生态模型的非线性奇摄动近似解

非线性奇摄动问题在国际学术界中是一个重要的研究对象。它涉及到许多学科。在一些生态现象中,原始的研究方法只是采取某些简单观察和统计数据来得到结论。但是它对生态现象的实质的研究达不到效果。近来在国际上提出了研究生态学的动力学方法,即人们首先把它归化为代表它的现象本质的微分方程的模型,然后用数学方法来求解对应的方程,最后研究关于生物和数学理论的动力学方面的规律。目前,非线性摄动问题已经被广泛地研究。许多学者已经研究了一些近似理论。近似求解方法已被发展,包括平均法,边界层法,匹配渐近展开和多尺度法等等。研究非线性广义Lotke-Volterra捕食-被捕食生态模型,一个简单而有效的摄动方法被应用到捕食-被捕食生态模型。提出了捕食-被捕食的一个模型,它是一个微分方程系统,并用小的正参数按幂级数展开未知函数,然后得到关于幂级数的系数的方程,并求出它们的解。于是利用摄动方法得到了原问题解的渐近展开式。得到了它是原模型解是一个好的近似的结论,它是一个解析展开式并且能保持其解析运算。最后,给出了一个对应的例子,它说明得到的解具有很好的精度。     

Three-species ecosystems in a solvable model

A detailed discussion of the three-species ecosystems is presented in an exactly solvable model with interactions of the Gompertz form. Three different possibilities, namely, a one-prey-two-predator system, a two-prey-one-predator system and a three-step prey-predator food chain are considered. These systems are studied not only when they include their basic prey-predator interactions, but also when various self-interactions as well as competition between like species, in different possible combinations, are included. It is then inferred, by obtaining and examining the exact solutions, as to when these systems possess stable equilibrium and when not, or when they are purely oscillatory, etc. We also study, within our model, the two-species versus three-species situation. It is seen that there are situations when the three-species system possesses stable equilibrium even under circumstances under which the corresponding two-species system is unstable. We also come across cases when the addition of the third species destroys the possibility of stable equilibrium which the initial two-species system possessed. Some other results also follow. Of particular interest is the one where the initial two-species system is purely oscillatory but the enlarged system, which is a three-step prey-predator chain, has the first and the last populations of the chain rising indefinitely and the middle population remains oscillatory. A comparison of our results with results of other authors, wherever possible, has also been made.     

Application of perturbation theory to the nonlinear Volterra-Gause-Witt model for prey-predator interaction

Krylov-Bogoliubov-Mitropolsky perturbation method was used to study the effect of nonlinearity in the Volterra-gause-Witt (VGW) model for a two species prey-predator system. The first order corrections to both the frequency of oscillation and the amplitude of the linearized system were computed. It was found that the basic qualitative features of the nonlinearity are exhibited by the first order result. We have also discussed the Lotka-Volterra problem which is a special case of VGW model.     

具性别偏食的二种群捕食者-食饵系统模型

本文根据生态学实验现象研究了一类二种群捕食者－食饵系统,其中捕食者对食饵有性别偏食情形。建立了相应的数学模型,并对该模型的有关性状进行了分析。     

Bifurcation analysis of a predator-prey model with predators using hawk and dove tactics

Most classical prey-predator models do not take into account the behavioural structure of the population. Usually, the predator and the prey populations are assumed to be homogeneous, i.e. all individuals behave in the same way. In this work, we shall take into account different tactics that predators can use for exploiting a common self-reproducing resource, the prey population. Predators fight together in order to keep or to have access to captured prey individuals. Individual predators can use two behavioural tactics when they encounter to dispute a prey, the classical hawk and dove tactics. We assume two different time scales. The fast time scale corresponds to the inter-specific searching and handling for the prey by the predators and the intra-specific fighting between the predators. The slow time scale corresponds to the (logistic) growth of the prey population and mortality of the predator. We take advantage of the two time scales to reduce the dimension of the model and to obtain an aggregated model that describes the dynamics of the total predator and prey densities at the slow time scale. We present the bifurcation analysis of the model and the effects of the different predator tactics on persistence and stability of the prey-predator community are discussed.     

Aggregation and emergence in hierarchically organized systems: population dynamics

The aim of this work is to present aggregation methods of hierarchically organized systems allowing one to replace the initial micro-system by a macro-system described by a few global variables. We also study the relations between the fast micro-dynamics and the slow macro-dynamics which can produce global properties. Emergence corresponds to a bottom-up coupling that is the result effected by a micro-level at a macro-level. As an example, we present prey-predator models with different time scales in an heterogeneous environment. A fast time scale is associated to the migration process on spatial patches and a slow time scale is associated to growth and interactions between the populations. Preys must go on spatial patches where resources are located and where predators can attack them. The efficiency of the predators to catch preys is patch dependent. Perturbation methods allow us to aggregate the initial system of differential equations for the patch sub-populations into a macro-system of two differential equations governing the total population densities. We study the case of density independent and density dependent migrations. In the latter case, we show that different functional responses can emerge in the macro prey-predator model as a result of the coupling between the slow and fast systems.     

Complexity in a prey-predator model with prey refuge and diffusion

Prey-predator interaction is one of the most commonly observed relationships in ecosystem. In the study of prey-predator models, it is frequently assumed that the changes in population densities are only time-dependent and the dynamics is generally represented by coupled nonlinear ordinary differential equations. In natural system, however, either prey or predator or both move from one place to another for various reasons. In such a case, their dynamic interaction depends both on time and space and requires coupled nonlinear partial differential equations for its dynamic representation. It is also well documented that prey refuges affect the interaction between prey and predator significantly. In this paper, we studied the dynamics of a diffusive prey-predator interaction with prey refuge and type III response function. We have considered both one and two dimensional diffusivity in the model system and presented different stability results under the assumptions that one or both species may be mobile or sedentary. Our results showed that the system may exhibit different spatiotemporal (non-Turing) patterns, like spiral waves, patchy structures, spot pattern, or even spatiotemporal chaos depending on the refuge availability and diffusion rate of species. Another interesting finding was that the dynamic complexity in a prey-predator model increases in case of mobile predator and sedentary prey compare to mobile prey and sedentary predator while refuge availability is varied.     

Order and flexibility in the motion of fish schools

The coexistence of order and flexibility in the motion of fish schools was studied by using a simple numerical model and a computer simulation. The numerical model is based on behavioral rules for individuals in the school by considering attraction, repulsion, and parallel-orientation behavior. Each individual follows the same rules and makes school movements. The simulation results show that school order and flexibility are affected by the number of neighbors interacting with an individual in the school and by the randomness of individual motion. Increase in the number of interacting neighbors leads to high order, especially when the number increases from a low value (between one and three). An optimal number of interacting neighbors exists that is relatively low (two or three) for high flexibility, indicating that a fish needs only to pay attention to a few neighbors to realize both order and flexibility. The low randomness of individual motion benefits both order and flexibility. These results indicate that schooling fish have evolved specialized ability for establishing both school order and flexibility.     

Global dynamics of microbial communities emerge from local interaction rules

Most microbes live in spatially structured communities (e.g., biofilms) in which they interact with their neighbors through the local exchange of diffusible molecules. To understand the functioning of these communities, it is essential to uncover how these local interactions shape community-level properties, such as the community composition, spatial arrangement, and growth rate. Here, we present a mathematical framework to derive community-level properties from the molecular mechanisms underlying the cell-cell interactions for systems consisting of two cell types. Our framework consists of two parts: a biophysical model to derive the local interaction rules (i.e. interaction range and strength) from the molecular parameters underlying the cell-cell interactions and a graph based model to derive the equilibrium properties of the community (i.e. composition, spatial arrangement, and growth rate) from these local interaction rules. Our framework shows that key molecular parameters underlying the cell-cell interactions (e.g., the uptake and leakage rates of molecules) determine community-level properties. We apply our model to mutualistic cross-feeding communities and show that spatial structure can be detrimental for these communities. Moreover, our model can qualitatively recapitulate the properties of an experimental microbial community. Our framework can be extended to a variety of systems of two interacting cell types, within and beyond the microbial world, and contributes to our understanding of how community-level properties emerge from microscopic interactions between cells.     

Maintenance of butterfly populations with all-female broods under recurrent extinction and recolonization

A theory is considered which can explain the apparent persistence of butterfly populations containing a special type of females producing all-female broods. Under normal circumstances it would be expected that these populations should become extinct, but this does not happen in nature. The theory is based on the possibility of a balance between populations dying out and habitats being recolonized. It is shown that a stable equilibrium can exist in a simple model leading to a set of differential equations and also in a more realistic simulation model. The first approach may have a bearing on certain prey-predator systems as well.     

Adaptation of interacting populations to the environmental temperature regime

The biological process of the passing of individuals from the active to the passive state when unfavourable environmental conditions occur has been formalized. This leads to a universal linear superstructing to nonlinear models of ecosystems. At certain rates of this jump, a stabilization of the oscillation instability is possible, and an increase in population vitality can occur. The adaptation processes were examined, and the areas of evolutionarily stable parameters for interacting population (competitors and antagonists) on regular temperature changes were found. A possibility of cyclic changes of parameters in the "prey-predator" system was shown.     

Increased corticosterone levels in mice subjected to the rat exposure test

In recent years, there has been a notable interest in studying prey-predator relationships to develop rodent-based models for the neurobehavioral aspects of stress and emotion. However, despite the growing use of transgenic mice and results showing important differences in the behavioral responses of rats and mice, little research has been conducted regarding the responses of mice to predators. The rat exposure test (RET), a recently developed and behaviorally validated prey-predator (mouse-rat)-based model, has proven to be a useful tool in evaluating the defensive responses of mice facing rats. To further validate the RET, we investigated the endocrine and behavioral responses of mice exposed to this apparatus. We first constructed a plasma corticosterone secretion curve in mice exposed to a rat or to an empty cage (control). Rat-exposed mice showed a pronounced rise in corticosterone levels that peaked 15 min from the beginning of the predator exposure. The corticosterone levels and behavioral responses of mice exposed to a rat or to a toy in the RET apparatus were then measured. We observed high plasma corticosterone levels along with clear avoidance behaviors represented by decreases in tunnel and surface area exploration and increases in risk assessment behaviors and freezing. This strongly suggests that the test elicits a repertoire of behavioral responses compatible with an aversion state and indicates that it is a promising model for the evaluation of prey-predator interactions. However, more physiological, neurochemical, and pharmacological studies are needed to further validate the test.     

Analytical ultracentrifugation as a tool for studying protein interactions

The last two decades have led to significant progress in the field of analytical ultracentrifugation driven by instrumental, theoretical, and computational methods. This review will highlight key developments in sedimentation equilibrium (SE) and sedimentation velocity (SV) analysis. For SE, this includes the analysis of tracer sedimentation equilibrium at high concentrations with strong thermodynamic non-ideality, and for ideally interacting systems, the development of strategies for the analysis of heterogeneous interactions towards global multi-signal and multi-speed SE analysis with implicit mass conservation. For SV, this includes the development and applications of numerical solutions of the Lamm equation, noise decomposition techniques enabling direct boundary fitting, diffusion deconvoluted sedimentation coefficient distributions, and multi-signal sedimentation coefficient distributions. Recently, effective particle theory has uncovered simple physical rules for the co-migration of rapidly exchanging systems of interacting components in SV. This has opened new possibilities for the robust interpretation of the boundary patterns of heterogeneous interacting systems. Together, these SE and SV techniques have led to new approaches to study macromolecular interactions across the entire spectrum of affinities, including both attractive and repulsive interactions, in both dilute and highly concentrated solutions, which can be applied to single-component solutions of self-associating proteins as well as the study of multi-protein complex formation in multi-component solutions.     

The role of behavioral dynamics in determining the patch distributions of interacting species

The effect of the behavioral dynamics of movement on the population dynamics of interacting species in multipatch systems is studied. The behavioral dynamics of habitat choice used in a range of previous models are reviewed. There is very limited empirical evidence for distinguishing between these different models, but they differ in important ways, and many lack properties that would guarantee stability of an ideal free distribution in a single-species system. The importance of finding out more about movement dynamics in multispecies systems is shown by an analysis of the effect of movement rules on the dynamics of a particular two-species-two-patch model of competition, where the population dynamical equilibrium in the absence of movement is often not a behavioral equilibrium in the presence of adaptive movement. The population dynamics of this system are explored for several different movement rules and different parameter values, producing a variety of outcomes. Other systems of interacting species that may lack a dynamically stable distribution among patches are discussed, and it is argued that such systems are not rare. The sensitivity of community properties to individual movement behavior in this and earlier studies argues that there is a great need for empirical investigation to determine the applicability of different models of the behavioral dynamics of habitat selection.     

Alcohol Impairs Predation Risk Response and Communication in Zebrafish

The effects of ethanol exposure on Danio rerio have been studied from the perspectives of developmental biology and behavior. However, little is known about the effects of ethanol on the prey-predator relationship and chemical communication of predation risk. Here, we showed that visual contact with a predator triggers stress axis activation in zebrafish. We also observed a typical stress response in zebrafish receiving water from these conspecifics, indicating that these fish chemically communicate predation risk. Our work is the first to demonstrate how alcohol effects this prey-predator interaction. We showed for the first time that alcohol exposure completely blocks stress axis activation in both fish seeing the predator and in fish that come in indirect contact with a predator by receiving water from these conspecifics. Together with other research results and with the translational relevance of this fish species, our data points to zebrafish as a promising animal model to study human alcoholism.