Prey-predator models in spatially heterogeneous environments

The effects of environmental heterogeneity on models of prey-predator systems are investigated. Refuge behaviour is found in a continuous gradually varying environment. In this situation we do not necessarily get oscillating population cycles. The stabilizing effect observed depends on environmental variation and is not produced by diffusion alone. Our conclusions are fairly independent of the details of the model.     

Understanding foraging behaviour in spatially heterogeneous environments

The role of stochasticity and spatial heterogeneity in foraging systems is investigated. We formulate a spatially explicit model which describes the behaviour of grazing animals in response to local information using simple stochastic rules. In particular the model reflects the biology in that decisions to move to a new location are based on visual assessment of the sward height in a surrounding neighbourhood, whilst the decision to graze the current location is based on the residual sward height and olfactory assessment of local faecal contamination. It is assumed that animals do not interact directly, but do so through modification of, and response to a common environment. Spatial heterogeneity is shown to have significant effects including reducing the equilibrium intake rate and increasing the optimal stocking density, and must therefore be taken into account by resource managers. We demonstrate the relationship between the stochastic spatial model and its non-spatial deterministic counterpart, and in the process derive a moment-closure approximation to the full process, which can be regarded as an intermediate, or pseudo-spatial model. The role of spatial heterogeneity is emphasized, and better understood by comparing the results obtained from each approach. The relative efficiency of random and directed searching behaviour in spatially heterogeneous environments is explored for both clean and contaminated pastures, and the impact of faecal avoidance behaviour assessed.     

Models of infectious diseases in spatially heterogeneous environments

Most models of dynamics of infectious diseases have assumed homogeneous mixing in the host population. However, it is increasingly recognized that heterogeneity can arise through many processes. It is then important to consider the existence of subpopulations of hosts, and that the contact rate within subpopulations is different than that between subpopulations. We study models with hosts distributed in subpopulations as a consequence of spatial partitioning. Two types of models are considered. In the first one there is direct transmission. The second one is a model of dynamics of a mosquito-borne disease, with indirect transmission, and applicable to malaria. The contact between subpopulations is achieved through the visits of hosts. Two types of visit are considered: a first one in which the visit time is independent of the distance travelled, and a second one in which visit time decreases with distance. There are two types of spatial arrangement: one dimensional, and two dimensional. Conditions for the establishment of the disease are obtained. Results indicate that the disease becomes established with greater difficulty when the degree of spatial partition increases, and when visit time decreases. In addition, when visit time decreases with distance, the establishment of the disease is more difficult when the spatial arrangement is one dimensional than when it is two dimensional. The results indicate the importance of knowing the spatial distribution and mobility patterns to understand the dynamics of infectious diseases. The consequences of these results for the design of public health policies are discussed.     

Stochastic population growth in spatially heterogeneous environments: the density-dependent case

Journal of Mathematical Biology - This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for...     

Endemic disease in environments with spatially heterogeneous host populations

The main interest in epidemic models stems from their use in uncovering certain qualitative features of epidemic processes. A deterministic model of a general epidemic in a population with an arbitrary number of separate population centers is presented. The mixing within each center is assumed to be homogeneous, and the usual threshold theorem holds for each population. The mixing between centers is nonhomogeneous. This model is used to identify the necessary and sufficient conditions under which a disease will become endemic in the general population when each population center is below the threshold required for establishment of the disease and does not mix with other centers. These conditions depend critically on the concavity of the infection rate function with respect to the length of exposure time. The application of these results to host-vector models is discussed.     

Evolution of RNA virus in spatially structured heterogeneous environments

A hallmark of the infectious cycle for many RNA viruses parasitizing multicellular hosts is the need to invade and successfully replicate in tissues that comprise a variety of cell types. Thus, multicellular hosts represent a heterogeneous environment to evolving viral populations. To understand viral adaptation to multicellular hosts, we took a double approach. First, we developed a mathematical model that served to make predictions concerning the dynamics of viral populations evolving in heterogeneous environments. Second, the predictions were tested by evolving vesicular stomatitis virus in vitro on a spatially structured environment formed by three different cell types. In the absence of gene flow, adaptation was tissue-specific, but fitness in all tissues decreased with migration rate. The performance in a given tissue was negatively correlated with its distance to the tissue hosting the population. This correlation decreased with migration rate.     

Intraguild predation with evolutionary dispersal in a spatially heterogeneous environment

Journal of Mathematical Biology - In many cases, the motility of species in a certain region can depend on the conditions of the local habitat, such as the availability of food and other resources...     

Population differences in the timing of diapause: adaptation in a spatially heterogeneous environment

Summary Populations of the planktonic copepod, Diaptomus sanguineus, live in permanent and temporary freshwater ponds in Rhode Island. All ponds in which they occur become uninhabitable at some time during the year, but the nature and timing of the harsh period varies both spatially and temporally. Females produce discrete clutches either of subitaneous eggs which hatch immediately or of diapausing eggs which hatch the following season. The two egg types show distinct chorion morphologies under transmission electron microscopy. In permanent ponds the copepods start making diapausing eggs in March, one month before rising water temperatures induce planktivorous sunfish to become active. In temporary ponds diapausing eggs are produced, in a complex pattern from May to July, before the water disappears in late summer or early fall.We investigated the spatial scale at which D. sanguineus is adapted to this complex environment. In a reciprocal transfer experiment between temporary and permanent bodies of water, female copepods placed in new ponds made subies of water, female copepods placed in new ponds made subitaneous and diapausing eggs in the same sequence as control females retained in their home ponds. The copepod populations enter diapause at times appropriate for the local habitat conditions they experience, but inappropriate for other, nearby ponds. Transplanted females were unable to sense a change in pond type or to adjust egg production accordingly. We conclude that D. sanguineus populations are adapted to the specific conditions of isolated ponds rather than to a broader geographical region containing several pond types.     

The evolution of life histories in spatially heterogeneous environments: Optimal reaction norms revisited

Summary Natural populations live in heterogeneous environments, where habitat variation drives the evolution of phenotypic plasticity. The key feature of population structure addressed in this paper is the net flow of individuals from source (good) to sink (poor) habitats. These movements make it necessary to calculate fitness across the full range of habitats encountered by the population, rather than independently for each habitat. As a consequence, the optimal phenotype in a given habitat not only depends on conditions there but is linked to the performance of individuals in other habitats. We generalize the Euler-Lotka equation to define fitness in a spatially heterogeneous environment in which individuals disperse among habitats as newborn and then stay in a given habitat for life. In this case, maximizing fitness (the rate of increase over all habitats) is equivalent to maximizing the reproductive value of newborn in each habitat but not to maximizing the rate of increase that would result if individuals in each habitat were an isolated population. The new equation can be used to find optimal reaction norms for life history traits, and examples are calculated for age at maturity and clutch size. In contrast to previous results, the optimal reaction norm differs from the line connecting local adaptations of isolated populations each living in only one habitat. Selection pressure is higher in good and frequent habitats than in poor and rare ones. A formula for the relative importance of these two factors allows predictions of the habitat in which the genetic variance about the optimal reaction norm should be smallest.     

Carbon dioxide fluxes in a spatially and temporally heterogeneous temperate grassland

Landscape position, grazing, and seasonal variation in precipitation and temperature create spatial and temporal variability in soil processes, and plant biomass and composition in grasslands. However, it is unclear how this variation in plant and soil properties affects carbon dioxide (CO₂) fluxes. The aim of this study is to explore the effect of grazing, topographic position, and seasonal variation in soil moisture and temperature on plant assimilation, shoot and soil respiration, and net ecosystem CO₂ exchange (NEE). Carbon dioxide fluxes, vegetation, and environmental variables were measured once a month inside and outside long-term ungulate exclosures in hilltop (dry) to slope bottom (mesic) grassland throughout the 2004 growing season in Yellowstone National Park. There was no difference in vegetation properties and CO₂ fluxes between the grazed and the ungrazed sites. The spatial and temporal variability in CO₂ fluxes were related to differences in aboveground biomass and total shoot nitrogen content, which were both related to variability in soil moisture. All sites were CO₂ sinks (NEE>0) for all our measurments taken throughout the growing season; but CO₂ fluxes were four- to fivefold higher at sites supporting the most aboveground biomass located at slope bottoms, compared to the sites with low biomass located at hilltops or slopes. The dry sites assimilated more CO₂ per gram aboveground biomass and stored proportionally more of the gross-assimilated CO₂ in the soil, compared to wet sites. These results indicate large spatio-temporal variability of CO₂ fluxes and suggest factors that control the variability in Yellowstone National Park.     

Using spatially explicit models to characterize foraging performance in heterogeneous landscapes

ABSTRACT The success of most foragers is constrained by limits to their sensory perception, memory, and locomotion. However, a general and quantitative understanding of how these constraints affect foraging benefits, and the trade-offs they imply for foraging strategies, is difficult to achieve. This article develops foraging performance statistics to assess constraints and define trade-offs for foragers using biased random walk behaviors, a widespread class of foraging strategies that includes area-restricted searches, kineses, and taxes. The statistics are expected payoff and expected travel time and assess two components of foraging performance: how effectively foragers distinguish between resource-poor and resourcerich parts of their environments and how quickly foragers in poor parts of the environment locate resource concentrations. These statistics provide a link between mechanistic models of individuals' movement and functional responses, population-level models of forager distributions in space and time, and foraging theory predictions of optimal forager distributions and criteria for abandoning resource patches. Application of the analysis to area-restricted search in coccinellid beetles suggests that the most essential aspect of these predators's foraging strategy is the "turning threshold," the prey density at which ladybirds switch from slow to rapid turning. This threshold effectively determines whether a forager exploits or abandons a resource concentration. Foraging is most effective when the threshold is tuned to match physiological or energetic requirements. These performance statistics also help anticipate and interpret the dynamics of complex spatially and temporally varying forager-resource systems.     

Scale-dependent responses in cabbage herbivores affect attack rates in spatially heterogeneous systems

Herbivorous insects face a dilemma when selecting suitable hosts in a complex environment, and their sensory capability may often reduce the female capacity for proper selection. As a consequence, eggs are often deposited on inferior hosts, affecting both insect and host plant fitness. We examined the attack rates of three cabbage herbivores in monocultures and biculture plots of different Brassica oleracea genotypes, with different spatial heterogeneity. The main goals of the study were to improve our understanding of the spatial scales involved in herbivore search processes and to examine the possibility of using spatial heterogeneity for manipulating pest attack rates in cabbage cropping systems. The results showed that the host selection behaviour of the small white butterfly (Pieris rapae) was strongly dependent on spatial heterogeneity. The difference in egg density between plant genotypes was larger when contrasting plants were growing in close proximity than in monoculture. This suggests that P. rapae is able to differentiate among genotypes from a small distance, while selection is compromised at larger spatial scales. The two other herbivores in the study (Mamestra brassicae and Delia radicum) did not respond to heterogeneity at any spatial scale, but showed a constant preference hierarchy. This suggests that host selection in these species occurs after direct plant contact. The difference in species’ responses to spatial heterogeneity has consequences both for selection gradients in natural communities and for the potential to reduce pest attack in polyculture systems.     

Synchronization of coupled biological oscillators under spatially heterogeneous environmental forcing

Spatially heterogeneous intensities of environmental signals are common in nature, being caused, e.g., by rugged or curved surfaces leading to varying angles of incidence and intensities. In this work, we perform numerical studies of one-dimensional arrays of coupled phase oscillators driven by a periodic signal with spatially heterogeneous amplitude, considering both random and gradual amplitude distributions of the driving. We compare the effects of global and next-neighbor interactions, respectively, on the mutual and forced synchronization in the array. Weak global coupling leads to full mutual synchronization for all studied driving configurations. The degree of external synchronization follows a majority rule, depending on the number of externally entrained oscillators in the uncoupled case. The effects of next-neighbor coupling depend on the spatial distribution of the driving amplitude. For random distributions, local interactions show the same qualitative effects as global coupling. In contrast, for gradual distributions and large driving heterogeneities, next-neighbor coupling is detrimental to both mutual and external synchronization. We discuss these observations with respect to fundamental aspects of heterogeneity and variability of dynamical systems, as well as the intercellular synchronization of circadian oscillators.     

Growth responses of some soil fungi to spatially heterogeneous nutrients

Abstract The natural nutritional environments of most fungi are spatially non-uniform, yet the majority of studies of fungal growth take no account of this fact. An experimental system is described which permits the growth responses of eucarpic fungi to heterogeneously distributed nutrient resources to be studied. The system comprises tesselations of agar tiles of contrasting nutrient status separated by air gaps. Growth responses in such systems of Alternaria alternata, Mucor sp., Phoma foveata , Rhizoctonia solani and Trichoderma viride are described. Generally, the growth of the fungi reflected the nutrient status of the underlying substrate. There was evidence for growth in low-nutrient tiles being greater when high-nutrient tiles were included in the tessellation. Reproductive structures tended to be formed only in low nutrient tiles with Trichoderma and Rhizoctonia and only high nutrient tiles with Alternaria . Growth responses of Rhizoctonia were strongly asymmetric in nutritionally symmetric, but heterogeneous, tesselations. The consequences of the observations for fungal growth in heterogeneous environments such as soil is discussed.     

Life-history evolution in spatially heterogeneous environments,with and without phenotypic plasticity

Summary The formulation of Kawecki and Stearns (1993) adapted for sexual populations is used to characterize lifehistory evolution in spatially heterogeneous environments comprising a number of distinct habitats. Three types of evolutionary outcome/optimal strategy are distinguished, appertaining to populations with phenotypic plasticity, populations without it (here called aplastic) and to populations that are reproductively isolated. In general plastic and isolated optima differ, but do not differ if none of the habitats provide source or sink populations. Plastic, aplastic and isolated optima are calculated and compared in three numerical examples representing trade-offs involving reproductive effort, egg size and defence. Locating the aplastic optimum involves numerical construction of a fitness landscape showing how allelic fitness depends on aplastic strategy and on the relative frequencies of the habitats. In all three examples the aplastic optimum lies between or almost between the plastic optima. In two cases the aplastic optimum switches abruptly between the plastic optima as the relative frequencies of the habitats change, and in the third case the switch is gradual. The abruptness or otherwise of the switch depends on the position and structure of the valleys in the fitness landscape and this in turn depends on how sharply the fitness peaks are differentiated.     

Endozoochorous seed dispersal by cattle and horse in a spatially heterogeneous landscape

Seed dispersal has become an important issue in plant ecology and restoration management. In this paper we examined dung germinating seed content and seed deposition patterns of horses (Shetland and Konik breeds) and Scottish Highland cattle grazing two coastal dune nature reserves. Two times 2.5 l of fresh dung from each type of herbivore were collected during seven sessions in the main fruiting season. Dung samples were placed under greenhouse conditions after drying and cooling. Animal defecation patterns were derived from a study of herbivore activities during 6 h observation sessions 8 times a month. One hundred and seventeen plant species i.e. 27% of all species occurring in the study area, were recorded as seedlings emerging from the dung samples. The most abundantly and frequently recorded plant species were Urtica dioica, Juncus spp. and different species of Poaceae and Caryophyllaceae. In general seedling density is high (1158 seedlings/dung sample). Seedling density and species richness were further analysed in order to detect temporal variability and possible animal and site related characteristics. Dung deposition patterns reflect a non-random use of habitats and hence a non-random seed deposition among habitats. Calculated seed deposition per square meter ranged from a few (<10 germinating seeds) to more than 100 in the most frequently selected habitats. From the herbivores’ selective habitat use and their estimated mean retention time we can further assume their ability for inter-habitat endozoochorous seed dispersal. This characteristic of large herbivores is further discussed in the light of nature management and restoration.Indra Lamoot is an aspirant of FWO-Flanders.