Effects of the probability of a predator catching prey on predator–prey system stability

To understand the effect of the probability of a predator catching prey, P_catch, on the stability of the predator–prey system, a spatially explicit lattice model consisting of predators, prey, and grass was constructed. The predators and prey randomly move on the lattice space, and the grass grows according to its growth probability. When a predator encounters prey, the predator eats the prey in accordance with the probability P_catch. When a prey encounters grass, the prey eats the grass. The predator and prey give birth to offspring according to a birth probability after eating prey or grass, respectively. When a predator or prey is initially introduced or newly born, its health state is set at a high given value. This health state decreases by one with every time step. When the state of an animal decreases to less than zero, the individual dies and is removed from the system. Population densities for predator and prey fluctuated significantly according to P_catch. System stability was characterized by the standard deviation ? of the fluctuation. The simulation results showed that ? for predators increased with an increase of P_catch; ? for prey reached a maximum at P_catch = 0.4; and ? for grass fluctuated little regardless of P_catch. These results were due to the tradeoff between P_catch and the predator–prey encounter rate, which represents the degree of interaction between predator and prey and the average population density, respectively.     

Analyzing the spatial dynamics of a prey–predator lattice model with social behavior

A lattice prey–predator model is studied. Transition rules applied sequentially describe processes such as reproduction, predation, and death of predators. The movement of predators is governed by a local particle swarm optimization algorithm, which causes the formation of swarms of predators that propagate through the lattice. Starting with a single predator in a lattice fully covered by preys, we observe a wavefront of predators invading the zones dominated by preys; subsequent fronts arise during the transient phase, where a monotonic approach to a fixed point is present. After the transient phase the system enters an oscillatory regime, where the amplitude of oscillations appears to be bounded but is difficult to predict. We observe qualitative similar behavior even for larger lattices. An empirical approach is used to determine the effects of the movement of predators on the temporal dynamics of the system. Our results show that the algorithm used to model the movement of predators increases the proficiency of predators.     

Ontogenetic shifts in a prey’s chemical defences influence feeding responses of a snake predator

Foraging theory suggests that predator responses to potential prey should be influenced by prey chemical defences, but the effects of ontogenetic variation in such defences on prey vulnerability to predators remain unclear. Cane toads (Rhinella marina) are toxic to anurophagous snakes, including the keelback (Tropidonophis mairii, a natricine colubrid that occurs within the toads' invasive range in Australia). Toxin levels and diversity change through toad ontogeny, decreasing from the egg stage to metamorphosis, then increasing in postmetamorphic toads. If the toxin content of a prey item influences predator responses, we predict that keelbacks should exhibit selective predation on toads close to metamorphosis. The results of our laboratory trials on adult (field-collected, and thus toad-experienced) and hatchling (laboratory-incubated, and thus toad-naive) keelbacks supported this prediction. The snakes selectively consumed later-stage rather than earlier-stage tadpoles, and earlier-stage rather than later-stage metamorphs. Our data are thus consistent with the hypothesis that ontogenetic changes in toxin content can affect individuals' vulnerability to predation.     

Existence and non-existence of spatial patterns in a ratio-dependent predator–prey model

In this paper, first we consider the global dynamics of a ratio-dependent predator–prey model with density dependent death rate for the predator species. Analytical conditions for local bifurcation and numerical investigations to identify the global bifurcations help us to prepare a complete bifurcation diagram for the concerned model. All possible phase portraits related to the stability and instability of the coexisting equilibria are also presented which are helpful to understand the global behaviour of the system around the coexisting steady-states. Next we extend the temporal model to a spatiotemporal model by incorporating diffusion terms in order to investigate the varieties of stationary and non-stationary spatial patterns generated to understand the effect of random movement of both the species within their two-dimensional habitat. We present the analytical results for the existence of globally stable homogeneous steady-state and non-existence of non-constant stationary states. Turing bifurcation diagram is prepared in two dimensional parametric space along with the identification of various spatial patterns produced by the model for parameter values inside the Turing domain. Extensive numerical simulations are performed for better understanding of the spatiotemporal dynamics. This work is an attempt to make a bridge between the theoretical results for the spatiotemporal model of interacting population and the spatial patterns obtained through numerical simulations for parameters within Turing and Turing–Hopf domain.     

Revealing the role of predator interference in a predator–prey system with disease in prey population

Predation on a species subjected to an infectious disease can affect both the infection level and the population dynamics. There is an ongoing debate about the act of managing disease in natural populations through predation. Recent theoretical and empirical evidence shows that predation on infected populations can have both positive and negative influences on disease in prey populations. Here, we present a predator–prey system where the prey population is subjected to an infectious disease to explore the impact of predator on disease dynamics. Specifically, we investigate how the interference among predators affects the dynamics and structure of the predator–prey community. We perform a detailed numerical bifurcation analysis and find an unusually large variety of complex dynamics, such as, bistability, torus and chaos, in the presence of predators. We show that, depending on the strength of interference among predators, predators enhance or control disease outbreaks and population persistence. Moreover, the presence of multistable regimes makes the system very sensitive to perturbations and facilitates a number of regime shifts. Since, the habitat structure and the choice of predators deeply influence the interference among predators, thus before applying predators to control disease in prey populations or applying predator control strategy for wildlife management, it is essential to carefully investigate how these predators interact with each other in that specific habitat; otherwise it may lead to ecological disaster.     

Analysis of a competitive prey–predator system with a prey refuge

Gauss's competitive exclusive principle states that two competing species having analogous environment cannot usually occupy the same space at a time but in order to exploit their common environment in a different manner, they can co-exist only when they are active in different times. On the other hand, several studies on predators in various natural and laboratory situations have shown that competitive coexistence can result from predation in a way by resisting any one prey species from becoming sufficiently abundant to outcompete other species such that the predator makes the coexistence possible. It has also been shown that the use of refuges by a fraction of the prey population exerts a stabilizing effect in the interacting population dynamics. Further, the field surveys in the Sundarban mangrove ecosystem reveal that two detritivorous fishes, viz. Liza parsia and Liza tade (prey population) coexist in nature with the presence of the predator fish population, viz. Lates calcarifer by using refuges.     

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.     

Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system

Mechanisms and scenarios of pattern formation in predator–prey systems have been a focus of many studies recently as they are thought to mimic the processes of ecological patterning in real-world ecosystems. Considerable work has been done with regards to both Turing and non-Turing patterns where the latter often appears to be chaotic. In particular, spatiotemporal chaos remains a controversial issue as it can have important implications for population dynamics. Most of the results, however, were obtained in terms of ‘traditional’ predator–prey models where the per capita predation rate depends on the prey density only. A relatively new family of ratio-dependent predator–prey models remains less studied and still poorly understood, especially when space is taken into account explicitly, in spite of their apparent ecological relevance. In this paper, we consider spatiotemporal pattern formation in a ratio-dependent predator–prey system. We show that the system can develop patterns both inside and outside of the Turing parameter domain. Contrary to widespread opinion, we show that the interaction between two different type of instability, such as the Turing–Hopf bifurcation, does not necessarily lead to the onset of chaos; on the contrary, the emerging patterns remain stationary and almost regular. Spatiotemporal chaos can only be observed for parameters well inside the Turing–Hopf domain. We then investigate the relative importance of these two instability types on the onset of chaos and show that, in a ratio-dependent predator–prey system, the Hopf bifurcation is indeed essential for the onset of chaos whilst the Turing instability is not.     

The world from a cat’s perspective – statistics of natural videos

The mammalian visual system is one of the most intensively investigated sensory systems. However, our knowledge of the typical input it is operating on is surprisingly limited. To address this issue, we seek to learn about the natural visual environment and the world as seen by a cat. With a CCD camera attached to their head, cats explore several outdoor environments and videos of natural stimuli are recorded from the animals perspective. The statistical analysis of these videos reveals several remarkable properties. First, we find an anisotropy of oriented contours with an enhanced occurrence of horizontal orientations, earlier described in the oblique effect as a predominance of the two cardinal orientations. Second, contrast is not elevated in the center of the images, suggesting different mechanisms of fixation point selection as compared to humans. Third, analyzing a sequence of images we find that the precise position of contours varies faster than their orientation. Finally, collinear contours prevail over parallel shifted contours, matching recent physiological and anatomical results. These findings demonstrate the rich structure of natural visual stimuli and its direct relation to extensively studied anatomical and physiological issues.     

Spatiotemporal behavior of a prey–predator system with a group defense for prey

Group defense is a strategy widely employed by various species. We consider the effect of grouping on population persistence when animals join together in herds in order to provide a self-defense from predators. In literature, group defense is usually addressed in terms of individual behavioral responses. In this paper, we consider an alternative ‘mean-field’ approach which uses prey and predator densities as the dynamical variables. The model is essentially a predator–prey system but with an unconventional parametrization for the predation term. We discuss the outcomes of the ecosystem dynamics in terms of persistence and prey survival. In the spatially distributed model some specific spatio-temporal features are discovered.     

On a predator–prey system of Gause type

In this paper a Gause type model of interactions between predator and prey population is considered. We deal with the sufficient condition due to Kuang and Freedman in the generalized form including a kind of weight function. In a previous paper we proved that the existence of such weight function implies the uniqueness of limit cycle. In the present paper we give a new condition equivalent to the existence of a weight function (Theorem 4.4). As a consequence of our result, it is shown that some simple qualitative properties of the trophic function and the prey isocline ensure the uniqueness of limit cycle.     

What makes a good neighborhood? Interaction of spatial scale and fruit density in the predator satiation dynamics of a masting juniper tree

Spatio-temporal variability in fruit production (masting) has been regarded as a key mechanism to increase plant fitness by reducing seed predation. However, considerably more effort has been devoted into understanding the consequences of temporal rather than spatial variations in fruit crop for plant fitness. In order to simultaneously evaluate both components, we quantify fruit production and pre-dispersal damage by three arthropod species (mites, chalcid wasps and moths) in the Spanish juniper (Juniperus thurifera) during 3 years in a spatially explicit context. Our aims were to assess (1) the interaction between fruit production and pre-dispersal fruit damage by arthropods, (2) the potential interference or competition between arthropods, and (3) the form of the phenotypic selection exerted by arthropods on fruit traits considering the spatial context. Arthropods damaged a substantial fraction of fruits produced by Spanish juniper with levels of damage showing sharp inter-annual variations. Fruit damage by mites was negatively related to yearly fruit crop and positively correlated at individual trees fruiting in consecutive years. Increased interspecific interference was an additional consequence of reduced fruit availability during small crop years. During a masting year, fruit damage by less mobile species such as mites was negatively affected by tree crop size, and no spatial structure was observed for mite damage. The incidence of chalcid wasps was low, so the spatial pattern of seed predation was unclear, and no preferences for fruit or seed traits were detected. Conversely, moths selected larger fruits and their incidence on trees was spatially aggregated up to 20 m, with predation levels being negatively affected by fruit abundance at the patch level, suggesting a positive density-dependent effect of neighbors on fruit output. These results highlight the importance of including the spatial component to understand complex species interactions at local scales.     

Host plant quality,spatial heterogeneity,and the stability of mite predator–prey dynamics

Population dynamics models suggest that both the over-all level of resource productivity and spatial variability in productivity can play important roles in community dynamics. Higher productivity environments are predicted to destabilize consumer–resource dynamics. Conversely, greater heterogeneity in resource productivity is expected to contribute to stability. Yet the importance of these two factors for the dynamics of arthropod communities has been largely overlooked. I manipulated nutrient availability for strawberry plants in a multi-patch experiment, and measured effects of overall plant quality and heterogeneity in plant quality on the stability of interactions between the phytophagous mite Tetranychus urticae and its predator Phytoseiulus persimilis. Plant size, leaf N content and T. urticae population growth increased monotonically with increasing soil nitrogen availability. This gradient in plant quality affected two correlates of mite population stability, population variability over time (i.e., coefficient of variation) and population persistence (i.e., proportion of plant patches colonized). However, the highest level of plant quality did not produce the least stable dynamics, which is inconsistent with the “paradox of enrichment”. Heterogeneity in plant productivity had modest effects on stability, with the only significant difference being less variable T. urticae densities in the heterogeneous compared to the corresponding homogeneous treatment. These results are generally congruent with metapopulation theory and other models for spatially segregated populations, which predict that stability should be governed largely by relative movement rates of predators and prey—rather than patch quality.     

Global analysis of a predator–prey model with variable predator search rate

We consider a modified Holling-type II predator–prey model, based on the premise that the search rate of predators is dependent on the prey density, rather than constant. A complete analysis of the global behavior of the model is presented, and shows that the model exhibits a dichotomy similar to the classical Holling-type II model: either the coexistence steady state is globally stable; or it is unstable, and then a unique, globally stable limit cycle exists. We discuss the similarities, but also important differences between our model and the Holling-type II model. The main differences are that: 1. The paradox of enrichment which always occurs in the Holling-type II model, does not always occur here, and 2. Even when the paradox of enrichment occurs, predators can adapt by lowering their search rate, and effectively stabilize the system.

     

Effects of perturbation on the predator–prey system in a heterogeneous landscape

Environmental perturbations occur in ecosystems as the result of disturbance, which is closely related to ecosystem stability and resilience. To understand how perturbations can affect ecosystems, we constructed a spatially explicit lattice model to simulate the integrative predator–prey–grass relationships. In this model, a predator (or prey) gives birth to offspring, according to a specific birth probability, when it is able to feed on prey (or grass). When a predator or prey animal was initially introduced or newly born, its health state was set at a given high value. This state decreased by 1 with each time step. When the state of an animal decreased to zero, the animal was considered dead and was removed from the system. In this model, the perturbation was defined as the sudden death of some portion of the population. The heterogeneous landscape was characterized by a parameter, H, which controlled the degree of heterogeneity. When H ≥ 0.6, the predator population size was positively influenced by the perturbation. However, the perturbation had little effect upon the population sizes of prey or grass, regardless of the value of H.     

Parametric analysis of the ratio-dependent predator–prey model

We present a complete parametric analysis of stability properties and dynamic regimes of an ODE model in which the functional response is a function of the ratio of prey and predator abundances. We show the existence of eight qualitatively different types of system behaviors realized for various parameter values. In particular, there exist areas of coexistence (which may be steady or oscillating), areas in which both populations become extinct, and areas of "conditional coexistence" depending on the initial values. One of the main mathematical features of ratio-dependent models, distinguishing this class from other predator-prey models, is that the Origin is a complicated equilibrium point, whose characteristics crucially determine the main properties of the model. This is the first demonstration of this phenomenon in an ecological model. The model is investigated with methods of the qualitative theory of ODEs and the theory of bifurcations. The biological relevance of the mathematical results is discussed both regarding conservation issues (for which coexistence is desired) and biological control (for which extinction is desired).     

Dynamics analysis of a predator–prey system with harvesting prey and disease in prey species

In this paper, a predator–prey system with harvesting prey and disease in prey species is given. In the absence of time delay, the existence and stability of all equilibria are investigated. In the presence of time delay, some sufficient conditions of the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analysing the corresponding characteristic equation, and the properties of Hopf bifurcation are given by using the normal form theory and centre manifold theorem. Furthermore, an optimal harvesting policy is investigated by applying the Pontryagin's Maximum Principle. Numerical simulations are performed to support our analytic results.     

Numerical exploration of the parameter plane in a discrete predator–prey model

We propose a variant of the discrete Lotka–Volterra model for predator–prey interactions. A detailed stability and numerical analysis of the model are presented to explore the long time behaviour as each of the control parameter is varied independently. We show how the condition for survival of the predator depends on the natural death rate of predator and the efficiency of predation. The model is found to support different dynamical regimes asymptotically including predator extinction, stable fixed point and limit cycle attractors for co-existence of predator and prey and more complex dynamics involving chaotic attractors. We are able to locate exactly the domain of chaos in the parameter plane using a dimensional analysis.     

Do vegetation patch spatial patterns disrupt the spatial organization of plant species?

The mechanisms that structure plant diversity and generate long-range correlated spatial patterns have important implications for the conservation of fragmented landscapes. The ability to disperse and persist influences a plant species’ capacity for spatial organization, which can play a critical role in structuring plant diversity in metacommunities. This study examined the spatial patterns of species diversity within a network of patches in Cabo de Gata Natural Park, southeastern Spain. The objectives were to understand how the spatial heterogeneity of species composition (beta diversity) varies in a structured landscape, and how the long-range spatial autocorrelation of plant species is affected by the spatial configuration of patches.The mechanisms underlying the spatial distribution of plants acted at two scales. Between patches, spatial variation in species distributions was greater than that expected based on spatial randomization, which indicated that movement among patches was restricted. Within patches, diffusion processes reduced spatial variability in species distributions, and the effect was more prominent in large patches. Small patch size negatively influenced the long-range spatial autocorrelation of characteristic species, whereas inter-patch distance had a stronger effect on species frequency than it had on the disruption of spatial organized patterns.The long-range spatial autocorrelation was evaluated based on the dispersal abilities of the species. Among the 106 species evaluated, 39% of the woody species, 17% of the forbs, and 12% of the grasses exhibited disrupted long-range spatial autocorrelation where patches were small. The species that are more vulnerable to the effects of fragmentation tended to be those that have restricted dispersal, such as those that have short-range dispersal (atelechoric), e.g., Phlomis purpurea, Cistus albidus, Teucrium pseudochamaepytis, Brachypodium retusum, and the ballistic species, Genista spartioides. Helianthemum almeriense is another vulnerable species that has actively restricted dispersal (antitelechory), which is common in arid regions. Wind dispersers such as Launaea lanifera were less vulnerable to the effects of fragmentation. Long-distance dispersers whose persistence depends on facilitative interactions with other individuals, e.g., allogamous species such as Thymus hyemalis, Ballota hirsuta, and Anthyllis cytisoides, exhibit disrupted long-range spatial autocorrelation when patch size is reduced.     

Spatial heterogeneity stabilizes predator–prey interactions at the microscale while patch connectivity controls their outcome

Natural landscapes are both fragmented and heterogeneous, affecting the distribution of organisms, and their interactions. While predation in homogeneous environments increases the probability of population extinction, fragmentation/heterogeneity promotes coexistence and enhances community stability as shown by experimentation with animals and microorganisms, and supported by theory. Patch connectivity can modulate such effects but how microbial predatory interactions are affected by water-driven connectivity is unknown. In soil, patch habitability by microorganisms, and their connectivity depend upon the water saturation degree (SD). Here, using the obligate bacterial predator Bdellovibrio bacteriovorus, and a Burkholderia prey, we show that soil spatial heterogeneity profoundly affects predatory dynamics, enhancing long-term co-existence of predator and prey in a SD-threshold dependent-manner. However, as patches and connectors cannot be distinguished in these soil matrices, metapopulations cannot be invoked to explain the dynamics of increased persistence. Using a set of experiments combined with statistical and physical models we demonstrate and quantify how under full connectivity, predation is independent of water content but depends on soil microstructure characteristics. In contrast, the SD below which predation is largely impaired corresponds to a threshold below which the water network collapses and water connectivity breaks down, preventing the bacteria to move within the soil matrix.