Pattern formation in prey-taxis systems

In this paper, we consider spatial predator–prey models with diffusion and prey-taxis. We investigate necessary conditions for pattern formation using a variety of non-linear functional responses, linear and non-linear predator death terms, linear and non-linear prey-taxis sensitivities, and logistic growth or growth with an Allee effect for the prey. We identify combinations of the above non-linearities that lead to spatial pattern formation and we give numerical examples. It turns out that prey-taxis stabilizes the system and for large prey-taxis sensitivity we do not observe pattern formation. We also study and find necessary conditions for global stability for a type I functional response, logistic growth for the prey, non-linear predator death terms, and non-linear prey-taxis sensitivity.     

Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system

We consider a continuous taxis-diffusion-reaction system of partial-differential equations describing spatiotemporal dynamics of a predator–prey system. The local kinetics of the system is defined by general Gause–Kolmogorov-type model. The predator ability to pursue the prey is modelled by the Patlak–Keller–Segel taxis model, assuming that movement velocities of predators are proportional to the gradients of specific cues emitted by prey (e.g., odour, pheromones, exometabolites). The linear stability analysis of the model showed that the non-trivial homogeneous stationary regime of the model becomes unstable with respect to small heterogeneous perturbations with increase of prey-taxis activity; an Andronov–Hopf bifurcation occurs in the system when the taxis coefficient of predator exceeds its critical bifurcation value that exists for all admissible values of model parameters. These findings generalize earlier results obtained for particular cases of the Gause–Kolmogorov-type model assuming logistic reproduction of the prey population and the Holling types I and II functional responses of the predator population. Numerical simulations with theta-logistic growth of the prey population and the Ivlev functional response of predators illustrate and support results of the analytical study.     

Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics

The dynamics of a simple prey-predator system is described by a system of two reaction- diffusion equations with biologically reasonable non-linearities (logistic growth of the prey, Holling type II functional response of the predator). We show that, when the local kinetics of the system is oscillatory, for a wide class of initial conditions the evolution of the system leads to the formation of a non-stationary irregular pattern corresponding to spatio-temporal chaos. The chaotic pattern first appears inside a sub-domain of the system. This sub-domain then steadily grows with time and, finally, the chaotic pattern invades the whole space, displacing the regular pattern.     

Density‐dependent and ‐independent effects on the joint use of space by predators and prey in terrestrial arthropod food‐webs

The spatial distribution of predators and their prey is affected by their joint use of space. While the formation of such spatial patterns may be driven by density‐dependent and ‐independent factors our knowledge on the contribution of different land‐use activities on the formation of spatial patterns between predators and prey remains very limited. Agriculture is one of the most prevailing land‐use activities with strong effects on invertebrate densities and structural habitat conditions. Here, we used replicated conventionally and organically managed winter wheat fields to investigate the effects of agricultural land‐use on the spatial patterns of generalist predators and decomposer prey. We then identified the explanatory power of density‐dependent (prey and predator activity density) and density‐independent (vegetation structure) predictors for the observed spatial patterns. Generalist predators were regularly distributed only in conventionally managed fields and this pattern intensified with decreasing Collembola prey availability and increasing spider activity density. Segregation between carabid and spider predators was strongest in fields with lowest wheat plant height, suggesting more intense intraguild interactions in structurally less complex habitats. Collembola were aggregated independent of management and aggregation was strongest in fields with highest Collembola and carabid activity density. Spiders and Collembola prey were associated, but higher aphid densities under conventional management weakened or interrupted this spatial relationship. We conclude that active control of crop plant physiognomy by growth hormones and herbicides in conventionally managed fields promotes predator–predator segregation and that a high availability of aphid prey seems to decouple predator–Collembola prey associations. Our results emphasise the need for a more mechanistic understanding of the effects of land‐use on the formation of spatial patterns and species interactions, especially under scenarios of environmental change and an ongoing loss of biodiversity.     

The influence of generalist predators in spatially extended predator–prey systems

The presence of generalist predators is known to have important ecological impacts in several fields. They have wide applicability in the field of biological control. However, their role in the spatial distribution of predator and prey populations is still not clear. In this paper, the spatial dynamics of a predator–prey system is investigated by considering two different types of generalist predators. In one case, it is considered that the predator population has an additional food source and can survive in the absence of the prey population. In the other case, the predator population is involved in intraguild predation, i.e., the source of the additional food of the predator coincides with the food source of the prey population and thus both prey and predator populations compete for the same resource. The conditions for linear stability and Turing instability are analyzed for both the cases. In the presence of generalist predators, the system shows different pattern formations and spatiotemporal chaos which has important implications for ecosystem functioning not only in terms of their predictability, but also in influencing species persistence and ecosystem stability in response to abrupt environmental changes. This study establishes the importance of the consideration of spatial dynamics while determining optimal strategies for biological control through generalist predators.     

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.     

Detecting predator–prey relationships in the sea

Understanding the strength and diversity of predator‐prey interactions among species is essential to understand ecosystem consequences of population‐level variation. Directly quantifying the predatory behaviour of wild fishes at large spatial scales (>100 m) in the open sea is fraught with difficulties. To date the only empirical approach has been to search for correlations in the abundance of predators and their putative prey. As an example we use this approach to search for predators of the keystone crown‐of‐thorns starfish. We show that this approach is unlikely to detect predator–prey linkages because the theoretical relationship is non‐linear, resulting in multiple possible prey responses for single given predator abundance. Instead we suggest some indication of the strength and ecosystem importance of a predator–prey relationship can be gained by using the abundance of both predators and their putative prey to parameterize functional response models.     

Monoclonal antibodies reveal changes in predator efficiency with prey spatial pattern

Spatially explicit predator–prey interactions can alter the predatory potential of natural enemies augmented through conservation biological control. To test hypotheses regarding such interactions and predatory efficiency, we used a combination of molecular techniques and mark–release–recapture to study the foraging behaviour of a generalist carabid predator, Poecilus cupreus , in response to spatial patterns of its cereal aphid prey ( Metapolophium dirhodum and Sitobion avenae ). Beetle and aphid numbers were measured across two grids of sampling locations, within which aphid spatial pattern had been manipulated to generate patchy and more homogenous distributions. Aphid consumption was measured by enzyme-linked immunosorbent assays (ELISA) of beetle gut contents, using an aphid-specific monoclonal antibody. Movement and distribution patterns suggest that P. cupreus does not aggregate at, nor instigate prey-taxis within, aphid patches. However, more than two-thirds of the 2169 P. cupreus tested by ELISA had consumed aphids and the proportion of beetles containing aphid proteins was positively related to aphid density. Against expectation, the proportion of predators feeding on aphids was greatest where prey were homogenously distributed, and this was attributed to the loss of partial refuges for prey in aphid patches. The functional value of this type of uniform foraging strategy is ideally suited to early colonization of the crop habitat, when aphid numbers are low, before populations build up and form strong spatial patterns.     

Avoiding toxic prey may promote harmful algal blooms

Blooms of freshwater cyanobacteria are a worldwide spread environmental issue. Despite toxin producing planktonic species are generally expected to be poor competitors for resources, dense blooms of toxic cyanobacteria, such as Microcystis, do often occur in nature. Experimental results suggest that the formation of such blooms is promoted by the predatory activity of zooplankton. In fact, such predator grazes on both the nontoxic and toxic species despite the toxicity of the latter actually inhibits its growth. We model this phenomenon through a Lotka–Volterra reaction–diffusion system. Our goal is to investigate the coupled role of toxicity and zooplankton's predation in the persistence of the toxic prey and to study the mechanisms behind the formation of spatially local toxic blooms. It is known that the classical Lotka-Volterra system consisting of one prey and one predator never exhibits pattern formation. In this paper, we show that the introduction of a toxic prey may destabilize the spatially homogeneous coexistence and trigger spatial pattern formation. We also show that local blooms more likely occur when predators avoid the toxic prey and when the strength of the toxicity is of an intermediate level.     

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.     

Metapopulation dynamics for spatially extended predator–prey interactions

Traditional metapopulation theory classifies a metapopulation as a spatially homogeneous population that persists on neighboring habitat patches. The fate of each population on a habitat patch is a function of a balance between births and deaths via establishment of new populations through migration to neighboring patches. In this study, we expand upon traditional metapopulation models by incorporating spatial heterogeneity into a previously studied two-patch nonlinear ordinary differential equation metapopulation model, in which the growth of a general prey species is logistic and growth of a general predator species displays a Holling type II functional response. The model described in this work assumes that migration by generalist predator and prey populations between habitat patches occurs via a migratory corridor. Thus, persistence of species is a function of local population dynamics and migration between spatially heterogeneous habitat patches. Numerical results generated by our model demonstrate that population densities exhibit periodic plane-wave phenomena, which appear to be functions of differences in migration rates between generalist predator and prey populations. We compare results generated from our model to results generated by similar, but less ecologically realistic work, and to observed population dynamics in natural metapopulations.     

Predator foraging success and habitat complexity: quantitative test of the threshold hypothesis

Summary Numerous studies have demonstrated a negative relationship between increasing habitat complexity and predator foraging success. Results from many of these studies suggest a non-linear relationship, and it has been hypothesised that some threshold level of complexity is required before foraging success is reduced significantly. We examined this hypothesis using largemouth bass (Micropterus salmoides) foraging on juvenile bluegill sunfish (Lepomis macrochirus) in various densities of artificial vegetation. Largemouth foraging success differed significantly among the densities of vegetation tested. Regression analysis revealed a non-linear relationship between increasing plant stem density and predator foraging success. Logistic analysis demonstrated a significant fit of our data to a logistic model, from which was calculated the threshold level of plant stem desity necessary to reduce predator foraging success. Studies with various prey species have shown selection by prey for more complex habitats as a refuge from predation. In this stydy, we also examined the effects of increasing habitat complexity (i.e. plant stem density) on choice of habitat by juvenile bluegills while avoiding predation. Plant stem density significantly effected choice of habitat as a refuge. The relationship between increasing habitat complexity and prey choice of habitat was found to be positive and non-linear. As with predator foraging success, logistic analysis demonstrated a significant fit of our data to a logistic model. Using this model we calculated the threshold level of habitat complexity required before prey select a habitat as a refuge. This density of vegetation proved to be considerably higher than that necessary to significantly reduce predator foraging success, indicating that bluegill select habitats safe from predation.Implications of these results and various factors which may affect the relationships described are discussed.     

The effects of habitat fragmentation on persistence of source-sink metapopulations in systems with predators and prey or apparent competitors.

We consider systems with one predator and one prey, or a common predator and two prey species (apparent competitors) in source and sink habitats. In both models, the predator species is vulnerable to extinction, if productivity in the source is insufficient to rescue demographically deficient sink populations. Conversely, in the model with two prey species, if the source is too rich, one of the prey species may be driven extinct by apparent competition, since the predator can maintain a large population because of the alternative prey. Increasing the rate of predator movement from the source population has opposite effects on prey and predator persistence. High emigration rate exposes the predator population to danger of extinction, reducing the number of individuals that breed and produce offspring in the source habitat. This may promote coexistence of prey by relaxing predation pressure and apparent competition between the two prey species. The number of sinks and spatial arrangement of patches, or connectivity between patches, also influence persistence of the species. More sinks favor the prey and fewer sinks are advantageous to the predator. A linear pattern with the source at one end is profitable for the predator, and a centrifugal pattern in which the source is surrounded by sinks is advantageous to the prey. When the dispersal rate is low, effects of the spatial structure may exceed those of the number of sinks. In brief, productivity in patches and patterns of connectivity between patches differentially influence persistence of populations in different trophic levels.     

Limit cycles in predator-prey models

The general model of interaction between one predator and one prey is studied. A unimodal function of rate of growth of the prey and concave down functional response of the predator is assumed. In this work it is shown that for a given natural number n there exist models possessing at least 2n + 1 limit cycles. It is also proved, applying the Hopf bifurcation theorem, that a model exists with a logistic growth rate of the prey and concave down functional response that has at least two limit cycles.     

The complex dynamics of a diffusive prey–predator model with an Allee effect in prey

This paper investigates complex dynamics of a predator–prey interaction model that incorporates: (a) an Allee effect in prey; (b) the Michaelis–Menten type functional response between prey and predator; and (c) diffusion in both prey and predator. We provide rigorous mathematical results of the proposed model including: (1) the stability of non-negative constant steady states; (2) sufficient conditions that lead to Hopf/Turing bifurcations; (3) a prior estimates of positive steady states; (4) the non-existence and existence of non-constant positive steady states when the model is under zero-flux boundary condition. We also perform completed analysis of the corresponding ODE model to obtain a better understanding on effects of diffusion on the stability. Our analytical results show that the small values of the ratio of the prey's diffusion rate to the predator's diffusion rate are more likely to destabilize the system, thus generate Hopf-bifurcation and Turing instability that can lead to different spatial patterns. Through numerical simulations, we observe that our model, with or without Allee effect, can exhibit extremely rich pattern formations that include but not limit to strips, spotted patterns, symmetric patterns. In addition, the strength of Allee effects also plays an important role in generating distinct spatial patterns.     

Bioeconomic harvesting of a prey–predator fishery

This paper deals with the problem of non-selective harvesting of a prey–predator system by using a reasonable catch-rate function instead of usual catch-per-unit-efforthypothesis. Here both the prey and the predator species obey the law of logistic growth. We have taken the predator functional response to prey density in such a form that each predator's functional response to the prey density approaches a constant as the prey population increases. Boundedness of the exploited system is examined. The existence of its steady states and their stability (local and global) are studied using Eigenvalue analysis. The existence of bionomic equilibria has been illustrated using a numerical example. The problem of determining the optimal harvesting policy is then solved by using Pontryagin's maximum principle.     

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.     

Habitat heterogeneity and mammalian predator–prey interactions

• 1 In predator–prey theory, habitat heterogeneity can affect the relationship between kill rates and prey or predator density through its effect on the predator's ability to search for, encounter, kill and consume its prey. Many studies of predator–prey interactions include the effect of spatial heterogeneity, but these are mostly based on species with restricted mobility or conducted in experimental settings.
• 2 Here, we aim to identify the patterns through which spatial heterogeneity affects predator–prey dynamics and to review the literature on the effect of spatial heterogeneity on predator–prey interactions in terrestrial mammalian systems, i.e. in freely moving species with high mobility, in non‐experimental settings. We also review current methodologies that allow the study of the predation process within a spatial context.
• 3 When the functional response includes the effect of spatial heterogeneity, it usually takes the form of predator‐dependent or ratio‐dependent models and has wide applicability.
• 4 The analysis of the predation process through its different stages may further contribute towards identifying the spatial scale of interest and the specific spatial mechanism affecting predator–prey interactions.
• 5 Analyzing the predation process based on the functional response theory, but separating the stages of predation and applying a multiscale approach, is likely to increase our insight into how spatial heterogeneity affects predator–prey dynamics. This may increase our ability to forecast the consequences of landscape transformations on predator–prey dynamics.

     

Modeling changes in predator functional response to prey across spatial scales

Extrapolation of predator functional responses from laboratory observations to the field is often necessary to predict predation rates and predator-prey dynamics at spatial and temporal scales that are difficult to observe directly. We use a spatially explicit individual-based model to explore mechanisms behind changes in functional responses when the scale of observation is increased. Model parameters were estimated from a predator-prey system consisting of the predator Delphastus catalinae (Coleoptera: Coccinellidae) and Bemisia tabaci biotype B (Hemiptera: Aleyrodidae) on tomato plants. The model explicitly incorporates prey and predator distributions within single plants, the search behavior of predators within plants, and the functional response to prey at the smallest scale of interaction (within leaflets) observed in the laboratory. Validation revealed that the model is useful in scaling up from laboratory observations to predation in whole tomato plants of varying sizes. Comparing predicted predation at the leaflet scale, as observed in laboratory experiments, with predicted predation on whole plants revealed that the predator functional response switches from type II within leaflets to type III within whole plants. We found that the magnitude of predation rates and the type of functional response at the whole plant scale are modulated by (1) the degree of alignment between predator and prey distributions and (2) predator foraging behavior, particularly the effect of area-concentrated search within plants when prey population density is relatively low. The experimental and modeling techniques we present could be applied to other systems in which active predators prey upon sessile or slow-moving species.