Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality?

A constant dilemma in theoretical ecology is knowing whether model predictions corrspond to real phenomena or whether they are artifacts of the modelling framework. The frequent absence of detailed ecological data against which models can be tested gives this issue particular importance. We address this question in the specific case of invasion in a predator-prey system with oscillatory population kinetics, in which both species exhibit local random movement. Given only these two basic qualitative features, we consider whether we can deduce any properties of the behaviour following invasion. To do this we study four different types of mathematical model, which have no formal relationship, but which all reflect our two qualitative ingredients. The models are: reaction-diffusion equations, coupled map lattices, deterministic cellular automata, and integrodifference equations. We present results of numerical simulations of the invasion of prey by predators for each model, and show that although there are certain differences, the main qualitative features of the behaviour behind invasion are the same for all the models. Specifically, there are either irregular spatiotemporal oscillations behind the invasion, or regular spatiotemporal oscillations with the form of a periodic travelling ''wake'', depending on parameter values. The observation of this behaviour in all types of model strongly suggests that it is a direct consequence of our basic qualitative assumptions, and as such is an ecological reality which will always occur behind invasion in actual oscillatory predator-prey systems.     

Absolute stability and dynamical stabilisation in predator-prey systems

Many ecological systems exhibit multi-year cycles. In such systems, invasions have a complicated spatiotemporal structure. In particular, it is common for unstable steady states to exist as long-term transients behind the invasion front, a phenomenon known as dynamical stabilisation. We combine absolute stability theory and computation to predict how the width of the stabilised region depends on parameter values. We develop our calculations in the context of a model for a cyclic predator-prey system, in which the invasion front and spatiotemporal oscillations of predators and prey are separated by a region in which the coexistence steady state is dynamically stabilised.     

Invasion success in communities with reciprocal intraguild predation depends on the stage structure of the resident population

The probability of individuals being targeted as prey often decreases as they grow in size. Such size‐dependent predation risk is very common in systems with intraguild predation (IGP), i.e. when predatory species interact through predation and competition. Theory on IGP predicts that community composition depends on productivity. When recently testing this prediction using a terrestrial experimental system consisting of two phytoseiid mite species, Iphiseius degenerans as the IG‐predator and Neoseiulus cucumeris as the IG‐prey, and pollen (Typha latifolia) as the shared resource, we could not find the predicted community shift. Instead, we observed that IG‐prey excluded IG‐predators when the initial IG‐prey/IG‐predator ratio was high, whereas the opposite held when the initial ratio was low, which is also not predicted by theory. We therefore hypothesized that the existence of vulnerable and invulnerable stages in the two populations could be an important driver of the community composition. To test this, we first demonstrate that IG‐prey adults indeed attacked IG‐predator juveniles in the presence of the shared resource. Second, we show that the invasion capacity of IG‐predators at high productivity levels indeed depended on the structure of resident IG‐prey populations. Third, we further confirmed our hypothesis by mimicking successive invasion events of IG‐predators into an established population of IG‐prey at high productivity levels, which consistently failed. Our results show that the interplay between stage structure of populations and reciprocal intraguild predation is decisive at determining the species composition of communities with intraguild predation.     

State-dependent risk-taking by predators in systems with defended prey

Even defended prey items may contain nutrients that can sustain predators in times of energetic need. Conversely, a well-fed predator might be expected to avoid attacking prey items that have a chance of being defended, particularly if there is an abundance of familiar palatable prey to support it. To further understand the implications of optimal state-dependent foraging behaviour by predators in systems that contain defended prey, I developed a stochastic dynamic programming model. This state-dependent approach formally accounts for the trade-off between avoiding starvation and minimising harm from attacking defended prey. It predicts that the mean attack probability of predators on defended models and their undefended mimics should decline in a sigmoidal fashion with increasing availability of alternative undefended prey, and that the foraging decisions of predators should in general be relatively insensitive to the probability that a potentially defended prey item is indeed defended. Some implications of these predictions are that conspicuous warning signals are more likely to evolve in systems that contain an abundance of alternative undefended prey, and that imperfect mimicry will provide almost complete protection to the mimic when predators are readily supported by alternative food sources. Somewhat surprisingly, increasing the density of nutritious undefended mimics while keeping the densities of all other prey types constant tended to decrease the attack rates of predators on encounter with mimics and their defended models. This increase in dietary conservatism arose because in these cases there would be more prey available to sustain the predator if it ever found itself critically low in energy.     

Predator overcomes the Allee effect due to indirect prey–taxis

A mathematical model for spatiotemporal dynamics of prey–predator system was studied by means of linear analysis and numerical simulations. The model is a system of PDEs of taxis–diffusion–reaction type, accounting for the ability of predators to detect the locations of higher prey density, which is formalized as indirect prey–taxis, according to hypothesis that the taxis stimulus is a substance being continuously emitted by the prey, diffusing in space and decaying with constant rate in time (e.g. odour, pheromone, exometabolit). The local interactions of the prey and predators are described by the classical Rosenzweig – MacArthur system, which is modified in order to take into account the Allee effect in the predator population. The boundary conditions determine the absence of fluxes of population densities and stimulus concentration through the habitat boundaries. The obtained results suggest that the prey–taxis activity of the predator can destabilize both the stationary and periodic spatially-homogeneous regimes of the species coexistence, causing emergence of various heterogeneous patterns. In particular, it is demonstrated that formation of local dense aggregations induced by prey–taxis allows the predators to overcome the Allee effect in its population growth, avoiding the extinction that occurs in the model in the absence of spatial effects.     

Coexistence of predator and prey in intraguild predation systems with ontogenetic niche shifts

In basic intraguild predation (IGP) systems, predators and prey also compete for a shared resource. Theory predicts that persistence of these systems is possible when intraguild prey is superior in competition and productivity is not too high. IGP often results from ontogenetic niche shifts, in which the diet of intraguild predators changes as a result of growth in body size (life-history omnivory). As a juvenile, a life-history omnivore competes with the species that becomes its prey later in life. Competition can hence limit growth of young predators, while adult predators can suppress consumers and therewith neutralize negative effects of competition. We formulate and analyze a stage-structured model that captures both basic IGP and life-history omnivory. The model predicts increasing coexistence of predators and consumers when resource use of stage-structured predators becomes more stage specific. This coexistence depends on adult predators requiring consumer biomass for reproduction and is less likely when consumers outcompete juvenile predators, in contrast to basic IGP. Therefore, coexistence occurs when predation structures the community and competition is negligible. Consequently, equilibrium patterns over productivity resemble those of three-species food chains. Life-history omnivory thus provides a mechanism that allows intraguild predators and prey to coexist over a wide range of resource productivity.     

Continuous Traveling Waves for Prey-Taxis

Spatially moving predators are often considered for biological control of invasive species. The question arises as to whether introduced predators are able to stop an advancing pest or foreign population. In recent studies of reaction–diffusion models, it has been shown that the prey invasion can only be stopped if the prey dynamics observes an Allee effect. In this paper, we include prey-taxis into the model. Prey-taxis describe the active movement of predators to regions of high prey density. This effect leads to the observation that predators are drawn away from the leading edge of a prey invasion where its density is low. This leads to counterintuitive result that prey-taxis can actually reduce the likelihood of effective biocontrol.     

The impact of variable stoichiometry on predator-prey interactions: a multinutrient approach

A model for prey and predators is formulated in which three essential nutrients can limit growth of both populations. Prey take up dissolved nutrients, while predators ingest prey, assimilate a fraction of ingested nutrients that depends on their current nutrient status, and recycle the balance. Although individuals are modeled as identical within populations, amounts of nutrients within individuals vary over time in both populations, with reproductive rates increasing with these amounts. Equilibria and their stability depend on nutrient supply conditions. When nutrient supply increases, unusual results can occur, such as a decrease of prey density. This phenomenon occurs if, with increasing nutrient, predators sequester rather than recycle nutrients. Furthermore, despite use of a linear functional response for predators, high nutrient supply can destabilize equilibria. Responses to nutrient supply depend on the balance between assimilation and recycling of nutrients by predators, which differs depending on the identity of the limiting nutrient. Applied to microbial ecosystems, the model predicts that the efficiency of organic carbon mineralization is reduced when supply of mineral nutrients is low and when equilibria are unstable. The extent to which predators recycle or sequester limiting nutrients for their prey is of critical importance for the stability of predator-prey systems and their response to enrichment.     

Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition

We study the influence of nonlocal intraspecies prey competition on the spatiotemporal patterns arising behind predator invasions in two oscillatory reaction–diffusion integro-differential models. We use three common types of integral kernels as well as develop a caricature system, to describe the influence of the standard deviation and kurtosis of the kernel function on the patterns observed. We find that nonlocal competition can destabilize the spatially homogeneous state behind the invasion and lead to the formation of complex spatiotemporal patterns, including stationary spatially periodic patterns, wave trains and irregular spatiotemporal oscillations. In addition, the caricature system illustrates how large standard deviation and low kurtosis facilitate the formation of these spatiotemporal patterns. This suggests that nonlocal competition may be an important mechanism underlying spatial pattern formation, particularly in systems where the competition between individuals varies over space in a platykurtic manner.     

Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition

We study the influence of nonlocal intraspecies prey competition on the spatiotemporal patterns arising behind predator invasions in two oscillatory reaction–diffusion integro-differential models. We use three common types of integral kernels as well as develop a caricature system, to describe the influence of the standard deviation and kurtosis of the kernel function on the patterns observed. We find that nonlocal competition can destabilize the spatially homogeneous state behind the invasion and lead to the formation of complex spatiotemporal patterns, including stationary spatially periodic patterns, wave trains and irregular spatiotemporal oscillations. In addition, the caricature system illustrates how large standard deviation and low kurtosis facilitate the formation of these spatiotemporal patterns. This suggests that nonlocal competition may be an important mechanism underlying spatial pattern formation, particularly in systems where the competition between individuals varies over space in a platykurtic manner.     

Alteration of sensory abilities regulates the spatial scale of nonlethal predator effects

Many studies have shown that nonlethal predator effects such as trait-mediated interactions (TMIs) can have significant impacts on the structure and function of communities, but the role that environmental conditions play in modulating the scale and magnitude of these effects has not been carefully investigated. TMIs occur when prey exhibit behavioral or physiological responses to predators and may be more prevalent when abiotic conditions increase prey reactions to consumers. The purpose of this study was to determine if turbulence would alter the distance over which prey in aquatic systems respond to chemical cues emitted by predators in nature, thus changing the scales over which nonlethal predator effects occur. Using hard clams and blue crabs as a model predator–prey system, we investigated the effects of turbulence on clam reactive distance to predatory blue crabs in the field. Results suggest that turbulence diminishes clam reactions to predators and that the environmental context must be considered when predicting the extent of indirect predator effects in natural systems.     

Modeling of oscillatory scenarios of the coexistence of competing populations

The scenarios of the formation of population distributions have been analyzed for a system of nonlinear reaction–diffusion–advection equations to describe the spatiotemporal distribution of predators and prey. The conditions that must be fulfilled for the model to belong to the class of cosymmetric systems were identified using an analytical approach. Computer simulations of a system with prey and two predators showed that the emergence of families of stationary distributions and oscillatory modes is possible when these conditions are met. The initial distributions of predators were shown to determine the character of the scenario (stationary or non-stationary) at certain combinations of parameters.     

利用元胞自动机研究一类捕食食饵模型中的斑块扩散现象

利用概率元胞自动机模型对空间隐式的、食饵具Allee效应的一类捕食食饵模型进行模拟,发现随着相关参数的变化,种群的空间扩散前沿由连续的扩散波逐渐转变为一种相互隔离的斑块向外扩散,这种斑块扩散现象与以往的扩散模式有所不同。研究结果表明：(1)在斑块扩散的情况下,相关参数的微小变化会导致种群灭绝或者形成连续的扩散波,即斑块扩散发生在种群趋于灭绝和连续扩散之间;(2)当种群的空间扩散方式为斑块扩散时,种群的扩散速度会变慢,与其他扩散方式下的速度有着明显的区别。该研究结果对生物入侵控制和外来物种监测有重要的启发和指导作用。     

Generation of periodic waves by landscape features in cyclic predator-prey systems

The vast majority of models for spatial dynamics of natural populations assume a homogeneous physical environment. However, in practice, dispersing organisms may encounter landscape features that significantly inhibit their movement. We use mathematical modelling to investigate the effect of such landscape features on cyclic predator-prey populations. We show that when appropriate boundary conditions are applied at the edge of the obstacle, a pattern of periodic travelling waves develops, moving out and away from the obstacle. Depending on the assumptions of the model, these waves can take the form of roughly circular 'target patterns' or spirals. This is, to our knowledge, a new mechanism for periodic-wave generation in ecological systems and our results suggest that it may apply quite generally not only to cyclic predator-prey interactions, but also to populations that oscillate for other reasons. In particular, we suggest that it may provide an explanation for the observed pattern of travelling waves in the densities of field voles (Microtus agrestis) in Kielder Forest (Scotland-England border) and of red grouse (Lagopus lagopus scoticus) on Kerloch Moor (northeast Scotland), which in both cases move orthogonally to any large-scale obstacles to movement. Moreover, given that such obstacles to movement are the rule rather than the exception in real-world environments, our results suggest that complex spatio-temporal patterns such as periodic travelling waves are likely to be much more common in the natural world than has previously been assumed.     

Evolutionarily stable defence and signalling of that defence

We examine the evolution and maintenance of defence and conspicuousness in prey species using a game theoretic model. In contrast to previous works, predators can raise as well as lower their attack probabilities as a consequence of encountering moderately defended prey. Our model predicts four distinct possibilities for evolutionarily stable strategies (ESSs) featuring maximum crypsis. Namely that such a solution can exist with (1) zero toxicity, (2) a non-zero but non-aversive level of toxicity, (3) a high, aversive level of toxicity or (4) that no such maximally cryptic solution exists. Maximally cryptic prey may still invest in toxins, because of the increased chance of surviving an attack (should they be discovered) that comes from having toxins. The toxin load of maximally cryptic prey may be sufficiently strong that the predators will find them aversive, and seek to avoid similar looking prey in future. However, this aversiveness does not always necessarily trigger aposematic signalling, and highly toxic prey can still be maximally cryptic, because the increased initial rate of attack from becoming more conspicuous is not necessarily always compensated for by increased avoidance of aversive prey by predators. In other circumstances, the optimal toxin load may be insufficient to generate aversion but still be non-zero (because it increases survival), and in yet other circumstances, it is optimal to make no investment in toxins at all. The model also predicts ESSs where the prey are highly defended and aversive and where this defence is advertised at a cost of increased conspicuousness to predators. In many circumstances there is an infinite array of these aposematic ESSs, where the precise appearance is unimportant as long as it is highly visible and shared by all members of the population. Yet another class of solutions is possible where there is strong between-individual variation in appearance between conspicuous, poorly defended prey.     

Global stability and optimisation of a general impulsive biological control model

An impulsive model of augmentative biological control consisting of a general continuous predator-prey model in ordinary differential equations, i.e. a meta-model, augmented by a discrete part describing periodic introductions of predators is considered. The existence of an invariant periodic solution that corresponds to prey eradication is shown and a condition ensuring its global asymptotic stability is given. An optimisation problem related to the preemptive use of augmentative biological control is then considered. It is assumed that the per time unit budget of biological control (i.e. the number of predators to be released) is fixed and the best deployment of this budget is sought in terms of release frequency. The cost function to be minimised is the time needed to reduce an unforeseen prey (pest) invasion occurring at a worst time instant under some harmless level. The analysis shows that the optimisation problem admits a countable infinite number of solutions. An argumentation considering the required robustness of the optimisation result with respect to the invasive prey population level and to the model parameters is then conducted. It is shown that the cost function is decreasing in the predator release frequency so that the best deployment of the biocontrol agents is to carry out as frequent introductions as possible.     

Identifying the ecological conditions that select for intermediate levels of aposematic signalling

Chemically defended species often have conspicuous signals that warn potential predators of these defences. Recent evidence suggests that some such aposematic prey are not as conspicuous as possible, even though increased conspicuousness would bring additional anti-predator benefits. Here we present a simple model to explore the generality of these observations. Our model predicts that optimal fitness will often be achieved at an intermediate level of conspicuousness and not simply by maximising conspicuousness. This comes about because of the ubiquitous trade-off that increased conspicuousness has an ecological cost in increasing the encounter rate with predators, as well as a benefit in terms of enhancing learned aversion by predators of defended prey. However, importantly, we also predict that a small deviation away from maximal crypsis generally causes a decrease in fitness, even if a larger deviation would lead to an intermediate level of conspicuousness that maximises fitness. Hence, further consideration of whether intermediate levels of aposematism are as common in nature as predicted in this model will require consideration of the underlying evolution of appearance, and the plausibility of evolution across the fitness trough, from maximal crypsis to an intermediate level of aposematism.     

Strategies of aphidophagous predators: lessons for modelling insect predator–prey dynamics

Both theoretical and empirical evidence indicate that in systems where insect predators have longer developmental times than their prey the predators have little impact on the abundance of their prey. In assessing the 'effectiveness' of a predator for biological control one should take into account that selection maximizes predator fitness, not its effctiveness as a biocontrol agent. Therefore, predators that have a long developmental time relative to their prey are unlikely to be the best biocontrol agents. If these results can be generalized to other predator–prey systems, then it is clear that an understanding of predator–prey dynamics can only be achieved by studying predators.     

Prey preference of top predators manipulates the functioning and stability of multi-trophic ecosystems

The relationship between biodiversity and ecosystem functioning, and the mechanisms underpinning the food web stability, have been intensively investigated in ecological research. The ubiquities of generalists in natural food webs and its important role in dictating these ecosystem properties have been generally recognized. However, how competition between multiple top predators shape these ecosystem properties and determine the success of invasive predators remain largely unexplored. Here, we use a well-developed food web model to investigate the effects of prey preference of top predators on ecosystem functioning and food web stability in both local and invasive conditions. We design several modeling scenarios to mimic combinations of different types of top predators (specialist/generalist) and their origins (local/invasive). Our model theoretically shows that lower exploitation competition for prey between top predators (with distinct prey preferences featured by higher attack rates) would be beneficial for the ecosystem functioning and food web stability. We also demonstrate that the success of top predator invasion depends on the prey preference of both local and invasive top predators. Sensitivity analysis on the model further supports our findings. Our results highlight the importance of prey preference of multiple top predators in manipulating the properties of multi-trophic ecosystems. Our findings may have important implications because the current ongoing global changes profoundly change the phenology of many biological systems and create trophic mismatch, which may manipulate prey preference of top predators and in turn deteriorate ecosystem functioning and food web stability.