Investigation on stability and controller design for singular bio-economic systems with stochastic fluctuations

In this paper, the finite-time stability and control of a kind of singular bio-economic systems with stochastic fluctuations are investigated. When economic profit is no longer a constant but a variable, the system shows distinct dynamic behavior. First, a singular system is proposed to describe the bio-economic system with stochastic fluctuation. Then a singular stochastic T-S fuzzy model is established based on T-S fuzzy system theory. Second, a sufficient condition is proposed to satisfy finite-time stochastic stability of bio-economic system. On this basis, a fuzzy state feedback controller is designed which can make corresponding closed-loop singular stochastic bio-economic system admissible in finite-time, and the states of the system can be driven to a bounded range through the management of the open resource. Finally, the validity of the results is verified through the numerical simulation.

Harvesting creates ecological traps: consequences of invisible mortality risks in predator-prey metacommunities

Models of two-patch predator-prey metacommunities are used to explore how the global predator population changes in response to additional mortality in one of the patches. This could describe the dynamics of a predator in an environment that includes a refuge area where that predator is protected and a spatially distinct ("risky") area where it is harvested. The predator's movement is based on its perceived fitness in the two patches, but the risk from the additional mortality is potentially undetectable; this often occurs when the mortality is from human harvesting or from a novel type of top predator. Increases in undetected mortality in the risky area can produce an abrupt collapse of either the refuge population or of the entire predator population when the mortality rate exceeds a threshold level. This is due to the attraction of the risky patch, which has abundant prey due to its high predator mortality. Extinction of the refuge predator population does not occur when the refuge patch has a higher maximum per capita predator growth rate than the exploited patch because the refuge is then more attractive when the predator is rare. The possibility of abrupt extinction of one or both patches from high densities in response to a small increase in harvest is often associated with alternative states. In such cases, large reductions in mortality may be needed to avoid extinction in a collapsing predator population, or to reestablish an extinct population. Our analysis provides a potential explanation for sudden collapses of harvested populations, and it argues for more consideration of adaptive movement in designing protected areas.

Explaining variation in the effect of floral density on pollinator visitation

Pollinator responses to floral density have important implications for plant biology. In particular, a decline in pollinator visitation at low density can cause an Allee effect (a positive relation of fitness to density) in the plant population, which heightens that population's vulnerability to extinction. Empiricists have reported a variety of relations between flower or plant density and pollinator visitation rates. Here I develop and test a model that provides explanations for this diversity. The model assumes that pollinators distribute themselves between a focal patch of flowers and the surrounding environment so as to maximize foraging success. The resulting relation of per-flower visitation rate to focal-patch floral density is nonlinear, with positive effects at low floral densities and weaker or negative effects at higher densities. The relation is influenced by floral density in the surrounding environment and traits of both the plants and their pollinators. In a field experiment, floral density of Holocarpha virgata ssp. virgata had a nonlinear effect on per-flower visitation that was largely consistent with the model's predictions. By producing testable hypotheses based on biologically reasonable assumptions, this model serves as a starting point for explaining an important facet of plant-pollinator mutualisms.

Spatial dynamics of Microtus vole populations in continuous and fragmented agricultural landscapes

Small mammal populations often exhibit large-scale spatial synchrony, which is purportedly caused by stochastic weather-related environmental perturbations, predation or dispersal. To elucidate the relative synchronizing effects of environmental perturbations from those of dispersal movements of small mammalian prey or their predators, we investigated the spatial dynamics of Microtus vole populations in two differently structured landscapes which experience similar patterns of weather and climatic conditions. Vole and predator abundances were monitored for three years on 28 agricultural field sites arranged into two 120-km-long transect lines in western Finland. Sites on one transect were interconnected by continuous agricultural farmland (continuous landscape), while sites on the other were isolated from one another to a varying degree by mainly forests (fragmented landscape). Vole populations exhibited large-scale (>120 km) spatial synchrony in fluctuations, which did not differ in degree between the landscapes or decline with increasing distance between trapping sites. However, spatial variation in vole population growth rates was higher in the fragmented than in the continuous landscape. Although vole-eating predators were more numerous in the continuous agricultural landscape than in the fragmented, our results suggest that predators do not exert a great influence on the degree of spatial synchrony of vole population fluctuations, but they may contribute to bringing out-of-phase prey patches towards a regional density level. The spatial dynamics of vole populations were similar in both fragmented and continuous landscapes despite inter-landscape differences in both predator abundance and possibilities of vole dispersal. This implies that the primary source of synchronization lies in a common weather-related environment.

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.

Habitat Choice in Predator‐Prey Systems: Spatial Instability due to Interacting Adaptive Movements

The role of habitat choice behavior in the dynamics of predator-prey systems is explored using simple mathematical models. The models assume a three-species food chain in which each population is distributed across two or more habitats. The predator and prey adjust their locations dynamically to maximize individual per capita growth, while the prey's resource has a low rate of random movement. The two consumer species have Type II functional responses. For many parameter sets, the populations cycle, with predator and prey "chasing" each other back and forth between habitats. The cycles are driven by the aggregation of prey, which is advantageous because the predator's saturating functional response induces a short-term positive density dependence in prey fitness. The advantage of aggregation in a patch is only temporary because resources are depleted and predators move to or reproduce faster in the habitat with the largest number of prey, perpetuating the cycle. Such spatial cycling can stabilize population densities and qualitatively change the responses of population densities to environmental perturbations. These models show that the coupled processes of moving to habitats with higher fitness in predator and prey may often fail to produce ideal free distributions across habitats.

Migration Dynamics for the Ideal Free Distribution

This article verifies that the ideal free distribution (IFD) is evolutionarily stable, provided the payoff in each patch decreases with an increasing number of individuals. General frequency-dependent models of migratory dynamics that differ in the degree of animal omniscience are then developed. These models do not exclude migration at the IFD where balanced dispersal emerges. It is shown that the population distribution converges to the IFD even when animals are nonideal (i.e., they do not know the quality of all patches). In particular, the IFD emerges when animals never migrate from patches with a higher payoff to patches with a lower payoff and when some animals always migrate to the best patch. It is shown that some random migration does not necessarily lead to undermatching, provided migration occurs at the IFD. The effect of population dynamics on the IFD (and vice versa) is analyzed. Without any migration, it is shown that population dynamics alone drive the population distribution to the IFD. If animal migration tends (for each fixed population size) to the IFD, then the combined migration-population dynamics evolve to the population IFD independent of the two timescales (i.e., behavioral vs. population).

Spatial synchrony in host-parasitoid models using aggregation of variables

We consider a host-parasitoid system with individuals moving on a square grid of patches. We study the effects of increasing movement frequency of hosts and parasitoids on the spatial dynamics of the system. We show that there exists a threshold value of movement frequency above which spatial synchrony occurs and the dynamics of the system can be described by an aggregated model governing the total population densities on the grid. Numerical simulations show that this threshold value is usually small. This allows using the aggregated model to make valid predictions about global host-parasitoid spatial dynamics.

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.

Global production increased by spatial heterogeneity in a population dynamics model

Spatial and temporal heterogeneity are often described as important factors having a strong impact on biodiversity. The effect of heterogeneity is in most cases analyzed by the response of biotic interactions such as competition of predation. It may also modify intrinsic population properties such as growth rate. Most of the studies are theoretic since it is often difficult to manipulate spatial heterogeneity in practice. Despite the large number of studies dealing with this topics, it is still difficult to understand how the heterogeneity affects populations dynamics. On the basis of a very simple model, this paper aims to explicitly provide a simple mechanism which can explain why spatial heterogeneity may be a favorable factor for production. We consider a two patch model and a logistic growth is assumed on each patch. A general condition on the migration rates and the local subpopulation growth rates is provided under which the total carrying capacity is higher than the sum of the local carrying capacities, which is not intuitive. As we illustrate, this result is robust under stochastic perturbations.

Impact of spatial heterogeneity on a predator–prey system dynamics

This paper deals with the study of a predator-prey model in a patchy environment. Prey individuals moves on two patches, one is a refuge and the second one contains predator individuals. The movements are assumed to be faster than growth and predator-prey interaction processes. Each patch is assumed to be homogeneous. The spatial heterogeneity is obtained by assuming that the demographic parameters (growth rates, predation rates and mortality rates) depend on the patches. On the predation patch, we use a Lotka-Volterra model. Since the movements are faster that the other processes, we may assume that the frequency of prey and predators become constant and we would get a global predator-prey model, which is shown to be a Lotka-Volterra one. However, this simplified model at the population level does not match the dynamics obtained with the complete initial model. We explain this phenomenom and we continue the analysis in order to give a two-dimensional predator-prey model that gives the same dynamics as that provided by the complete initial one. We use this simplified model to study the impact of spatial heterogeneity and movements on the system stability. This analysis shows that there is a globally asymptotically stable equilibrium in the positive quadrant, i.e. the spatial heterogeneity stabilizes the equilibrium.

Integrative biology: linking levels of organization

Biological systems are composed of different levels of organization. Usually, one considers the atomic, molecular, cellular, individual, population, community and ecosystem levels. These levels of organization also correspond to different levels of observation of the system, from microscopic to macroscopic, i.e., to different time and space scales. The more microscopic the level is, the faster the time scale and the smaller the space scale are. The dynamics of the complete system is the result of the coupled dynamical processes that take place in each of its levels of organization at different time scales. Variables aggregation methods take advantage of these different time scales to reduce the dimension of mathematical models such as a system of ordinary differential equations. We are going to study the dynamics of a system which is hierarchically organized in the sense that it is composed of groups of elements that can be themselves divided into further smaller sub-groups and so on. The hierarchical structure of the system results from the fact that the intra-group interactions are assumed to be larger than inter-group ones. We present aggregation methods that allow one to build a reduced model that governs a few global variables at the slow time scale.

Migration frequency and the persistence of host-parasitoid interactions

This paper analyses the effect of migration frequency on the stability and persistence of a host-parasitoid system in a two-patch environment. The hosts and parasitoids are allowed to move from one patch to the other a certain number of times within a generation. When this number is low, i.e. when the time-scales associated with migration and demography are of the same order, host-parasitoid interactions are usually not persistent. When this number is high, however, persistence is more likely. Moreover, in this situation, aggregation methods can be used to simplify the proposed initial model into an aggregated model describing the dynamics of both the total host and parasitoid populations. Analysis of the aggregated model shows that the system reaches a stable steady state for some regions of the parameter domain. Persistence occurs when the movement of the parasitoids is asymmetrical, i.e. they move preferentially to one of the two patches. We show that the growth rate of the host population is a key parameter in determining which migration strategies of the parasitoids lead to persistent host-parasitoid interactions.

Landscape effects on temporal and spatial properties of vole population fluctuations

Populations of northern small rodents have previously been observed to fluctuate in spatial synchrony over distances ranging from tens to hundreds of kilometers between sites. It has been suggested that this phenomenon is caused by common environmental perturbations, mobile predators or dispersal movements. However, very little focus has been given to how the physical properties of the geographic area over which synchrony occurs, such as landscape composition and climate, affect spatial population dynamics. This study reports on the spatial and temporal properties of vole population fluctuations in two areas of western Finland: one composed of large interconnected areas of agricultural farmland interspersed by forests and the other highly dominated by forest areas, containing more isolated patches of agricultural land. Furthermore, the more agricultural area exhibits somewhat milder winters with less snow than the forested area. We found the amplitude of vole cycles to be essentially the same in the two areas, suggesting that the relative amount of predation on small rodents by generalist versus specialist predators is similar in both areas. No seasonal differences in the timing of synchronization were observable for Microtus voles, whereas bank vole populations in field habitats appeared to become synchronized primarily during winter. Microtus populations in field habitats exhibited smaller spatial variation and a higher degree of synchrony in the more continuous agricultural landscape than in the forest-dominated landscape. We suggest that this inter-areal difference is due to differences in the degree of inter-patch connectivity, with predators and dispersal acting as the primary synchronizing agents. Bank vole populations in field habitats were more synchronized within the forest-dominated landscape, most likely reflecting the suitability of the inter-patch matrix and the possibility of dispersal. Our study clearly indicates that landscape composition needs to be taken into account when describing the spatial properties of small rodent population dynamics.

Host patch selection induced by parasitism: basic reproduction ratio r(0) and optimal virulence

The effects of parasites on the behavior of their hosts are well documented. For example, parasites may affect the habitat selection of the host individual. We used variables aggregation methods to investigate the way in which parasites affect the spatial pattern of susceptible hosts. We developed a simple epidemiological model, taking into account both the reproduction processes of hosts (density-dependent birth and death) and infection, considered separately on two different patches, and the migration of susceptible hosts between these two patches. We used the complete model of three equations to generate an aggregated model describing the dynamics of the combined susceptible and infected host populations. We obtained the basic reproduction ratio (R(0)) from the aggregated model, and then studied the effect of the migratory behavior of susceptible hosts on the ability of the parasite to invade the system. We also used the basic reproduction ratio to investigate the evolution of parasite virulence in relation to the migration decisions of susceptible hosts. We found that host investment in avoidance of the infected patch leads to an increase in optimal virulence if host investment is costly.

A predator-prey model with predators using hawk and dove tactics

In this work we present a predator-prey model that incorporates individual behavior of the predators. A classical Lotka-Volterra model with self-limiting prey describes the predator-prey interaction. Predator individuals can use two behavioral tactics to dispute a prey when they meet, the classical hawk and dove tactics. Each individual can use both tactics along its life. The predator behavioral change is described by means of a game dynamic model based upon the replicator equations, where the gain depends on prey density. We assume that the demographic process, predator-prey interactions, acts at a slow time scale in comparison with the evolution of the behavior of the predator population. The existence of two time scales allows studying the complete system from a reduced one, which describes the dynamics of the total predator and prey densities at the slow time scale. The aim of this work is to study the effects of individual predator behavior on the dynamics of the predator-prey system. The main conclusion that emerges from this study is the existence of a relationship between prey density and the strategy adopted by predators: aggressive behavior is connected to high prey and low predator densities, whereas a polymorphism dove-hawk is found at low prey and high predator densities.

Swarming strategies for cooperative species

In this paper, we propose a model to investigate the relative efficiency of simple swarming strategies based on the interplay between spontaneous and recruitment-based emigration. We conduct a dynamical study of the model which combines inverse density dependence, saturation effects and induced vs. diffusion-like population transfer. The influence of the most relevant parameters is explored on a systematic basis, and transition values for which qualitative changes occur in the system's behaviour are given. The model is then used to study colonization of a multiple sites environment, as well as confrontation between species featuring different swarming strategies. Simulation results indicate that cooperative organisms should have an interest in evolving recruitment-based emigration. The corresponding population transfer patterns prove more efficient in invading new territories, eliminating competitors in the process. We suggest that this advantage could have promoted a simple form of coordinated swarming in species featuring a primitive type of cooperation.

Methods of aggregation of variables in population dynamics

In ecology, we are faced with modelling complex systems involving many variables corresponding to interacting populations structured in different compartmental classes, ages and spatial patches. Models that incorporate such a variety of aspects would lead to systems of equations with many variables and parameters. Mathematical analysis of these models would, in general, be impossible. In many real cases, the dynamics of the system corresponds to two or more time scales. For example, individual decisions can be rapid in comparison to growth of the populations. In that case, it is possible to perform aggregation methods that allow one to build a reduced model that governs the dynamics of a lower dimensional system, at a slow time scale. In this article, we present a review of aggregation methods for time continuous systems as well as for discrete models. We also present applications in population dynamics. A first example concerns a continuous time model of a single population distributed on a system of two connected patches (a logistic source and a sink), by fast migration. It is shown that under a certain condition, the total equilibrium population can be larger than the carrying capacity of the logistic source. A second example concerns a discrete model of a population distributed on two patches, still a source and a sink, connected by fast migration. The use of aggregation methods permits us to conclude that density-dependent migration can stabilize the total population.

Emergence of individual behaviour at the population level. Effects of density-dependent migration on population dynamics

The aim of this work is to study the effects of different individual behaviours on the overall growth of a spatially distributed population. The population can grow on two spatial patches, a source and a sink, that are connected by migrations. Two time scales are involved in the dynamics, a fast one corresponding to migrations and a slow one associated with the local growth on each patch. Different scenarios of density-dependent migration are proposed and their effects on the population growth are investigated. A general discussion on the use of aggregation methods for the study of integration of different ecological levels is proposed.