A functional response model of a predator population foraging in a patchy habitat

1. Functional response models (e.g. Holling's disc equation) that do not take the spatial distributions of prey and predators into account are likely to produce biased estimates of predation rates. 2. To investigate the consequences of ignoring prey distribution and predator aggregation, a general analytical model of a predator population occupying a patchy environment with a single species of prey is developed. 3. The model includes the density and the spatial distribution of the prey population, the aggregative response of the predators and their mutual interference. 4. The model provides explicit solutions to a number of scenarios that can be independently combined: the prey has an even, random or clumped distribution, and the predators show a convex, sigmoid, linear or no aggregative response. 5. The model is parameterized with data from an acarine predator-prey system consisting of Phytoseiulus persimis and Tetranychus urticae inhabiting greenhouse cucumbers. 6. The model fits empirical data quite well and much better than if prey and predators were assumed to be evenly distributed among patches, or if the predators were distributed independently of the prey. 7. The analyses show that if the predators do not show an aggregative response it will always be an advantage to the prey to adopt a patchy distribution. On the other hand, if the predators are capable of responding to the distribution of prey, then it will be an advantage to the prey to be evenly distributed when its density is low and switch to a more patchy distribution when its density increases. The effect of mutual interference is negligible unless predator density is very high. 8. The model shows that prey patchiness and predator aggregation in combination can change the functional response at the population level from type II to type III, indicating that these factors may contribute to stabilization of predator-prey dynamics.     

An epidemic model in a patchy environment

An epidemic model is proposed to describe the dynamics of disease spread among patches due to population dispersal. We establish a threshold above which the disease is uniformly persistent and below which disease-free equilibrium is locally attractive, and globally attractive when both susceptible and infective individuals in each patch have the same dispersal rate. Two examples are given to illustrate that the population dispersal plays an important role for the disease spread. The first one shows that the population dispersal can intensify the disease spread if the reproduction number for one patch is large, and can reduce the disease spread if the reproduction numbers for all patches are suitable and the population dispersal rate is strong. The second example indicates that a population dispersal results in the spread of the disease in all patches, even though the disease can not spread in each isolated patch.     

Towards a cognitive niche: divergent foraging strategies resulting from limited cognitive ability of foraging herbivores in a spatially complex environment.

A model was developed to explain one mechanism whereby differential optimal foraging strategies can occur between species as a result of selection for competition avoidance. This is the primary requirement for niche differentiation to evolve without a difference in the underlying foraging ability or morphology. The model used an individual-based patch choice mechanism, whereby herbivores move from patch to patch seeking food with the highest nutrient intake characteristics. The choice of patch was governed by a parameter, mu, which determined to what extent information in the landscape at different distances from the herbivore was used by it to make foraging decisions. A genetic algorithm was used to optimise the value, mu, in a complex landscape. The value of mu quickly converged to a single value with stabilising selection occurring when there was only a single species foraging. When there was a competing species with a fixed value of mu, the value of mu evolved to be above or below the mean for the single species mean depending on whether the value of mu for the competitor was below, or above the single-species mean, respectively. This was indicative of niche segregation. However mu tended to vary unstably over time when allowed to vary simultaneously in both species, although there was evidence for interaction between the two values. These results indicate that there can be a competitive advantage in choosing a cognitive strategy that is complementary to that used by other species.     

On optimal diet in a patchy environment

Optimal foraging theory has dealt with the following questions independently: (1) On what prey types should an individual predator feed (optimal diet)? (2) How long should a predator stay in each patch if prey is patchily distributed (optimal allocation of time to patches) ? This paper explores optimal foraging in patches containing several different kinds of prey. Results obtained by simulation show that deviations from recent predictions are to be expected, particularly for long interpatch travel times and rapid depletion of profitable prey types. In these situations the tactics of feeding as either specialist or as a generalist can be inferior to a tactic which starts as a specialist and then expands the diet after some time in the patch. Furthermore, predators should not necessarily stay longer in a patch if interpatch travel time increases. Some experimental tests of these new predictions are proposed.     

The Ross-Macdonald model in a patchy environment

We generalize to n patches the Ross-Macdonald model which describes the dynamics of malaria. We incorporate in our model the fact that some patches can be vector free. We assume that the hosts can migrate between patches, but not the vectors. The susceptible and infectious individuals have the same dispersal rate. We compute the basic reproduction ratio R(0). We prove that if R(0)1, then the disease-free equilibrium is globally asymptotically stable. When R(0)>1, we prove that there exists a unique endemic equilibrium, which is globally asymptotically stable on the biological domain minus the disease-free equilibrium.     

Oscillations in a patchy environment disease model

For a single patch SIRS model with a period of immunity of fixed length, recruitment-death demographics, disease related deaths and mass action incidence, the basic reproduction number R(0) is identified. It is shown that the disease-free equilibrium is globally asymptotically stable if R(0)<1. For R(0)>1, local stability of the endemic equilibrium and Hopf bifurcation analysis about this equilibrium are carried out. Moreover, a practical numerical approach to locate the bifurcation values for a characteristic equation with delay-dependent coefficients is provided. For a two patch SIRS model with travel, it is shown that there are several threshold quantities determining its dynamic behavior and that travel can reduce oscillations in both patches; travel may enhance oscillations in both patches; or travel can switch oscillations from one patch to another.     

Orientation and foraging movements in a patchy environment by the ant Serrastruma lujae (formicidae-myrmicinae)

Serrastruma lujae ants individually search for collembolans in the leaf litter of humid tropical forests. During the dry season, collembolans aggregate in wet patches randomly scattered in the dry litter where numerous single workers come hunting from their nest. We simulated this situation in the laboratory, and observed that workers seem to be able to use the humidity gradient direction to efficiently orient themselves towards a wet patch. Once the patch has been reached, they exhibit area-concentrated searching,consisting, in particular, of adopting a high sinuosity and a low speed. After capturing a collembolan, the ants return to their nest along nearly straight paths. This ability may rely on a spatial memory of the nest location by means of a path-integration process. In the absence of prey, however, various behaviours were observed after an unsuccessful search. Comparisons between these data and the results obtained with a homogeneously wet environment simulating the rainy season situation showed that these ants do not simply respond to the humidity level but are also sensitive to the degree of patchiness of their environment. They can therefore be said to be able to adapt suitably to the considerable climatic changes they encounter during the year.     

Transmission dynamics for vector-borne diseases in a patchy environment

In this paper, a mathematical model is derived to describe the transmission and spread of vector-borne diseases over a patchy environment. The model incorporates into the classic Ross–MacDonald model two factors: disease latencies in both hosts and vectors, and dispersal of hosts between patches. The basic reproduction number \(\mathcal{R }_0\) is identified by the theory of the next generation operator for structured disease models. The dynamics of the model is investigated in terms of \(\mathcal{R }_0\) . It is shown that the disease free equilibrium is asymptotically stable if \(\mathcal{R }_0<1\) , and it is unstable if \(\mathcal{R }_0>1\) ; in the latter case, the disease is endemic in the sense that the variables for the infected compartments are uniformly persistent. For the case of two patches, more explicit formulas for \(\mathcal{R }_0\) are derived by which, impacts of the dispersal rates on disease dynamics are also explored. Some numerical computations for \(\mathcal{R }_0\) in terms of dispersal rates are performed which show visually that the impacts could be very complicated: in certain range of the parameters, \(\mathcal{R }_0\) is increasing with respect to a dispersal rate while in some other range, it can be decreasing with respect to the same dispersal rate. The results can be useful to health organizations at various levels for setting guidelines or making policies for travels, as far as malaria epidemics is concerned.     

Basic reproduction ratio for a fishery model in a patchy environment

We present a dynamical model of a multi-site fishery. The fish stock is located on a discrete set of fish habitats where it is catched by the fishing fleet. We assume that fishes remain on fishing habitats while the fishing vessels can move at a fast time scale to visit the different fishing sites. We use the existence of two time scales to reduce the dimension of the model : we build an aggregated model considering the habitat fish densities and the total fishing effort. We explore a regulation procedure, which imposes an average residence time in patches. Several equilibria exist, a Fishery Free Equilibria (FFEs) as well as a Sustainable Fishery Equilibria (SFEs). We show that the dynamics depends on a threshold which is similar to a basic reproduction ratio for the fishery. When the basic reproduction ratio is less or equal to 1, one of the FFEs is globally asymptotically stable (GAS), otherwise one of the SFEs is GAS.     

Feeding of larval blue whiting and Atlantic mackerel: a comparison of foraging strategies

Fish larvae employ different feeding strategies depending on area and season of spawning and hatching of larvae. Feeding and growth of larvae of blue whiting Micromesistius poutassou and mackerel Scomber scombrus from Porcupine Bank and the Celtic Shelf Break, west of Ireland, were compared based on prey concentrations in the environment and larval feeding behaviour. Both species were adapted to different environmental conditions. The mesopelagic blue whiting spawned in oceanic water that was well mixed. It was characterized by low production and low prey densities with minimum prey densities <1.0 organism 1⁻¹. Larvae of the Atlantic mackerel hatched later in the season in more productive water that was well stratified. Prey densities in the mackerel environment reached up to 1001⁻¹. Blue whiting larvae displayed a rather random distribution in the water column. Mackerel larvae <7 mm standard length ( L _s) were concentrated above the thermocline, while larvae >7 mm traversed the thermocline into deeper layers. Mackerel larvae >5mm L _s displayed marked cannibalism, exceeding 70%. Daily ration calculated on the basis of gut contents was rather low in both species: between 2.6 and 5.0% in blue whiting, but only 0.6 to 5.4% in mackerel. The results are discussed in relation to the respective environment both species encounter during their early larval life.     

Root foraging for patchy resources in eight herbaceous plant species

The root foraging strategy of a plant species can be characterized by measuring foraging scale, precision, and rate. Trade-offs among these traits have been predicted to contribute to coexistence of competitors. We tested for trade-offs among root foraging scale (total root mass and length of structural roots), precision (ln-ratio of root lengths in resource-rich and resource-poor patches), and rate (days required for roots to reach a resource-rich patch, or growth rate of roots within a resource-rich patch) in eight co-occurring species. We found that root foraging scale and precision were positively correlated, as were foraging scale and the rate of reaching patches. High relative growth rate of a species did not contribute to greater scale, precision, or rate of root foraging. Introduced species had greater foraging scale, precision, and rate than native species. The positive correlations between foraging scale and foraging precision and rate may give larger species a disproportionate advantage in competition for patchy soil resources, leading to size asymmetric competition below ground.     

Sex allocation in a patchy environment: a marginal value theorem

The ESS sex allocation when male/female fitnesses vary with patch type is a set of values which either equalizes the marginal values of the male/female fitness tradeoffs, or are pure sexes. This is shown for a hermaphrodite; the result is then generalized to other sex allocation cases.     

Differential predation drives overyielding of prey species in a patchy environment

In environments characterized by regional heterogeneity among patches, competitor diversity can enhance ecosystem functions such as biomass production. Studies that have addressed the strength of diversity effects in heterogeneous environments have primarily considered a patchy distribution of resources. However, in many systems, top–down effects influence competitor productivity and composition. We use a three‐trophic level consumer–resource model to ask how differential responses to predation influence consumer diversity effects at two scales; 1) in patches with and without predator populations, and 2) at a ‘regional’ scale, consisting of one patch with‐ and one patch without a predator population. At the local scale, the strength and direction of consumer diversity effects depended on the strength of the differential response to predation. Positive or negative influences of consumer richness on equilibrium consumer biomass were the result of a selection effect of diversity. At the regional scale, we observed transgressive overyielding driven by a positive complementarity effect for parameters that define a strong differential response to predation. Given the prevalence of spatially and temporally heterogeneous top–down effects on competitor composition in many ecosystems and trophic levels, we advocate consideration of differential predation as an important step towards incorporating realistic trophic complexity into diversity–function studies.     

Seed dispersal in a patchy environment with global age dependence

A model of seed population dynamics proposed by S. A. Levin, A. Hastings, and D. Cohen is presented and analyzed. With the environment considered as a mosaic of patches, patch age is used along with time as an independent variable. Local dynamics depend not only on the local state, but also on the global environment via dispersal modelled by an integral over all patch ages. Basic technical properties of the time varying solutions are examined; necessary and sufficient conditions for nontrivial steady states are given; and general sufficient conditions for global asymptotic stability of these steady states are established. Primary tools of analysis include a hybrid Picard iteration, fixed point methods, monotonicity of solution structure, and upper and lower solutions for differential equations.This work was supported in part by National Science Foundation Grants MCS-7903497 and MCS-790349701     

Dietary and foraging strategies of baboons.

As large-bodied savannah primates, baboons have long been of special interest to students of human evolution: many different populations have been studied and dietary comparisons among them are becoming possible. Baboons' foraging strategies can be shown to combine high degrees of flexibility and breadth with selectivity. In this paper we develop and test multivariate models of the basis of diet selection for populations of montane and savannah baboons. Food selection is positively related to protein and lipid content and negatively to fibre, phenolics and alkaloids. Seasonal changes in dietary criteria predicted by these rules are tested and confirmed. Although nutritional bottlenecks occur at intervals, a comparison between long-term nutrient intakes in four different populations indicates convergence on lower degrees of variation than exist in superficial foodstuff profiles.     

Scaling the effects of predation and disturbance in a patchy environment

The effects of hydraulic disturbances on the impact of two predatory benthic invertebrates on their prey were examined in a stream at two distinct spatial scales. At the scale of small habitat patches (0.0625 m²), hydraulic patch type was an important determinant of the microdistribution of prey and predators. Prey abundances were similar across all patch types at baseflow, but local densities were higher in patches identified as low-flow refugia after periods of high and fluctuating flow. The microdistribution pattern of predatory larvae of a caddisfly, Plectrocnemia conspersa, was similar to that of its prey, whereas predatory larvae of an alderfly, Sialis fuliginosa, did not shift their microdistribution significantly with discharge and were always most abundant in lowflow refugia. There was little evidence of an aggregative response of predators with prey, even though both predators and prey are mobile. Both predator species showed similar patch-specific patterns of per capita consumption rates: uniform consumption rates across hydraulic patch types at low and moderate flows, but highest in flow refugia during high flows. Species-specific patterns, however, were apparent in the magnitude and direction of differences between consumption rates during disturbance events, and in comparable patches at base flow: At high flow, consumption rates for P. conspersa were exaggerated (3.9 times higher) in flow refugia but at par in other patches; for S. fuliginosa they were at par in flow refugia but reduced in other patches (up to 3.3. times lower). These differences may be related to species-specific foraging behaviours (search vs ambush predators) and the influence of prey movements on feeding success. Using the patch-scale results only, it is difficult to predict the effects of physical disturbance on predation intensity at the larger scales of whole habitats, populations or communities. At the large scale (>200 m²), net predator impacts were estimated over the stream reach, using a spatially explicit model that accounts, in an additive way, for habitat heterogeneity and patch-specific responses of predators and prey. The relationship between predator impact over the whole reach and hydraulic disturbance differed for the two predators. The predator impact of S. fuliginosa decreased with increasing hydraulic disturbance, as predicted by the harsh-benign hypothesis. There was no directional trend for P. conspersa, however, and maximum predator impact may occur at intermediate disturbance levels. For the prey community in this stream, predation pressure from S. fuliginosa appears to fluctuate directly with the discharge hydrograph, whereas predation from P. conspersa may be more persistent. Flow refugia may play a dual role in the sructure of stream communities by preventing catastrophic mortality of animals (predators and prey) from physical forces during disturbances, and by maintaining (or perhaps increasing) predation pressure. Summing the effects of species interactions in small habitat patches to the larger scale of a whole stream reach indicates that the scale of approach influences the observed patterns and their implied underlying process.     

Inverse density dependent parasitism in a patchy environment: a laboratory system

Abstract. 1. Two species of parasitoids (Anisopteromalus calandrae (Howard) and Heterospilus prosopidis Vier) attacking the bruchid beetle, Callosobruchus chinensis (L.), show marked inverse density dependent relationships between per cent parasitism and host density per patch.
2. These patterns are well described quantitatively using data on the spatial distribution of searching time by the parasitoids and their attack rates on patches of different host density.
3. A model of optimal foraging predicts just the opposite (i.e. density dependent) patterns of parasitism.
4. Both density dependent and inversely density dependent spatial patterns of parasitism can be explained mechanistically in terms of (a) the allocation of searching time in patches of different host density and (b) the maximum attack rate per parasitoid that constrains the extent of host exploitation within a patch.