Simple Predator-prey Model Research Articles

Stationary and oscillatory localized patterns in ratio-dependent predator–prey systems

Abstract Investigations are undertaken into simple predator–prey models with rational interaction terms in one and two spatial dimensions. Focusing on a case with linear interaction and saturation, an analysis for long domains in 1D is undertaken using ideas from spatial dynamics. In the limit that prey diffuses much more slowly than predator, the Turing bifurcation is found to be subcritical, which gives rise to localized patterns within a Pomeau pinning parameter region. Parameter regions for localized patterns and isolated spots are delineated. For a realistic range of parameters, a temporal Hopf bifurcation of the balanced equilibrium state occurs within the localized-pattern region. Detailed spectral computations and numerical simulations reveal how the Hopf bifurcation is inherited by the localized structures at nearby parameter values, giving rise to both temporally periodic and chaotic localized patterns. Simulation results in 2D confirm the onset of complex spatio-temporal patterns within the corresponding parameter regions. The generality of the results is confirmed by showing qualitatively the same bifurcation structure within a similar model with quadratic interaction and saturation. The implications for ecology are briefly discussed.

Barrier-mediated predator-prey dynamics

The survival chance of a prey chased by a predator depends not only on their relative speeds but importantly also on the local environment they have to face. Here, we propose a simple predator-prey model for a situation in which both the escaping prey and the chasing predator have to surmount an energetic barrier. Different barrier-assisted states of catching or final escaping are classified and suitable scaling laws separating these two states are derived. We discuss the effect of fluctuations on the catching times and determine states in which catching or escaping is more likely. We further identify trapping or escaping states which are determined by hydrodynamics and chemotactic interactions. Our results are of importance for both microbes and self-propelled unanimate microparticles following each other by non-reciprocal interactions in inhomogeneous landscapes.

Inferring ecosystem networks as information flows

The detection of causal interactions is of great importance when inferring complex ecosystem functional and structural networks for basic and applied research. Convergent cross mapping (CCM) based on nonlinear state-space reconstruction made substantial progress about network inference by measuring how well historical values of one variable can reliably estimate states of other variables. Here we investigate the ability of a developed optimal information flow (OIF) ecosystem model to infer bidirectional causality and compare that to CCM. Results from synthetic datasets generated by a simple predator-prey model, data of a real-world sardine-anchovy-temperature system and of a multispecies fish ecosystem highlight that the proposed OIF performs better than CCM to predict population and community patterns. Specifically, OIF provides a larger gradient of inferred interactions, higher point-value accuracy and smaller fluctuations of interactions and alpha-diversity including their characteristic time delays. We propose an optimal threshold on inferred interactions that maximize accuracy in predicting fluctuations of effective alpha-diversity, defined as the count of model-inferred interacting species. Overall OIF outperforms all other models in assessing predictive causality (also in terms of computational complexity) due to the explicit consideration of synchronization, divergence and diversity of events that define model sensitivity, uncertainty and complexity. Thus, OIF offers a broad ecological information by extracting predictive causal networks of complex ecosystems from time-series data in the space-time continuum. The accurate inference of species interactions at any biological scale of organization is highly valuable because it allows to predict biodiversity changes, for instance as a function of climate and other anthropogenic stressors. This has practical implications for defining optimal ecosystem management and design, such as fish stock prioritization and delineation of marine protected areas based on derived collective multispecies assembly. OIF can be applied to any complex system and used for model evaluation and design where causality should be considered as non-linear predictability of diverse events of populations or communities.

Chaotic dynamics in a simple predator-prey model with discrete delay

<p style='text-indent:20px;'>A discrete delay is included to model the time between the capture of the prey and its conversion to viable biomass in the simplest classical Gause type predator-prey model that has equilibrium dynamics without delay. As the delay increases from zero, the coexistence equilibrium undergoes a supercritical Hopf bifurcation, two saddle-node bifurcations of limit cycles, and a cascade of period doublings, eventually leading to chaos. The resulting periodic orbits and the strange attractor resemble their counterparts for the Mackey-Glass equation. Due to the global stability of the system without delay, this complicated dynamics can be solely attributed to the introduction of the delay. Since many models include predator-prey like interactions as submodels, this study emphasizes the importance of understanding the implications of overlooking delay in such models on the reliability of the model-based predictions, especially since temperature is known to have an effect on the length of certain delays.

Dynamics of a Predator–Prey Model with Distributed Delay to Represent the Conversion Process or Maturation

Distributed delay is included in a simple predator–prey model in the prey-to-predator biomass conversion term. The delayed term includes a delay-dependent “discount” factor that ensures the predators that do not survive the delay interval, do not contribute to growth of the predator population. A simple model was chosen so that without delay all solutions converge to a globally asymptotically stable equilibrium in order to show the possible effects of delay on the dynamics. If the co-existence equilibrium does not exist, the dynamics of the system is identical to its non-delayed analog. However, with delay, there is a delay-dependent threshold for the existence of the co-existence equilibrium. When the co-existence equilibrium exists, unlike the dynamics of the model without delay, a much wider range of dynamics is possible, including a strange attractor and bi-stability, although the system is uniformly persistent. A bifurcation theory approach is taken, using both the mean delay and the predator death rate as bifurcation parameters. We consider the gamma and the uniform distributions as delay kernels and show that the “discounting” term ensures that the Hopf bifurcations occur in pairs, as was observed in the analogous system with discrete delay (i.e., using the Dirac delta distribution). We show that there are certain features common to all distributions, although the model with different kernels can have a significantly different range of dynamics. In particular, the number of bi-stabilities, the sequence of bifurcations, the criticality of the Hopf bifurcations, and the size of the stability regions can differ. Also, the width of the interval over which the delay history is nonzero seems to have a significant effect on the range of dynamics. Thus, ignoring the delay and/or not choosing the right delay kernel might result in inaccurate modelling predictions.

Fisher information calculation in a complex ecological model: An optimal control-based approach

A new approach to the calculation of Fisher Information in a complex dynamic system is proposed in this work. The numerical strategy involves the successive solution of a series of optimal control problems to achieve the simultaneous estimation of the time cycle, as a fundamental parameter in the calculation of Fisher Information, and a set of Fisher Information average values, as well as the dynamic optimization of the system. A sequence of optimal control problems are solved first for estimating the time cycle, which is required to calculate the Fisher Information of the system. Then, the last optimization problem is solved to obtain not only the optimal profiles for the state and control variables, but also a set of time averaged Fisher Information values; such averaged values can be used to identify changes on the dynamic order of the system. Two illustrative examples are studied: a simple predator-prey Lotka-Volterra model is solved to explain the application of the numerical strategy for the estimation of the time cycle and time averaged Fisher Information values, and a proposed thirteen compartment socio-economic-ecological model, solved after the simple model to show the effectiveness of the approach to detect regime changes in a more complex system. The results of the simple model show a good approximation to the estimation of the time cycle after local minimum and maximum optimal points are obtained, since the system shows a regular cyclic behavior. For the larger model, the results confirm the effectiveness of the proposed optimal control based approach in more complex systems and the benefits of using Fisher Information average values to analyze the behavior of the system, while also identifying the policies that optimize the mass interactions among the compartments.

Evaluation of pest control efficiencies for different banker plant systems with a simple predator–prey model

AbstractThe banker plant system has been introduced for the biological control of various pest species in Japanese greenhouses. With the banker plant system, non‐crop plants infested with a host insect (a non‐commercial crop pest) are placed in the greenhouse to provide alternative resources for the parasitoids or predators. We want to evaluate the effectiveness for controlling pests on the crop in a quantitative way by immigrating predators from the banker plant. Therefore, we developed a simple model for the interaction of the pest and predator in the crop and included the banker plant only as a source for predators. For three different pest‐predator systems we parameterised the model and used these models to predict under what conditions biological control in a banker plant system is successful. We defined successful as keeping the pest below the economic injury level of the crop estimated from damage analysis. Because the crop is mostly grown during a period that lasts less than a year our analysis should not only focus on the equilibrium dynamics. In contrast, it should also focus on the transient dynamics. Our main analytical result, from the equilibrium analysis, is that for successful control the maximum lifetime consumption of immigrating predators should exceed the daily prey growth at half the value of the maximum consumption rate. For practical purpose this translates into the fact that the immigration of predators at a low initial pest density is crucial for successful control.

Evolutionary fields can explain patterns of high-dimensional complexity in ecology.

One of the properties that make ecological systems so unique is the range of complex behavioral patterns that can be exhibited by even the simplest communities with only a few species. Much of this complexity is commonly attributed to stochastic factors that have very high-degrees of freedom. Orthodox study of the evolution of these simple networks has generally been limited in its ability to explain complexity, since it restricts evolutionary adaptation to an inertia-free process with few degrees of freedom in which only gradual, moderately complex behaviors are possible. We propose a model inspired by particle-mediated field phenomena in classical physics in combination with fundamental concepts in adaptation, which suggests that small but high-dimensional chaotic dynamics near to the adaptive trait optimum could help explain complex properties shared by most ecological datasets, such as aperiodicity and pink, fractal noise spectra. By examining a simple predator-prey model and appealing to real ecological data, we show that this type of complexity could be easily confused for or confounded by stochasticity, especially when spurred on or amplified by stochastic factors that share variational and spectral properties with the underlying dynamics.

Dynamical analysis of a model of social behavior: Criminal vs non-criminal population

In this paper, we construct a model motivated by the well known predator-prey model to study the interaction between criminal population and non-criminal population. Our aim is to study various possibilities of interactions between them. First we model it using simple predator-prey model, then we modify it by considering the logistic growth of non-criminal population. We clearly deduce that the model with logistic growth is better than classical one. More precisely, the role of carrying capacity on the dynamics of criminal minded population is discussed. Further, we incorporate law enforcement term in the model and study its effect. The result obtained suggest that by incorporating enforcement law, the criminal population reduces from the very beginning, which resembles with real life situation. Our result indicates that the criminal minded population exist as long as coefficient of enforcement lc does not cross a threshold value and after this value the criminal minded population extinct. In addition, we also discuss the occurrence of saddle-node bifurcation in case of model system with law enforcement. Numerical examples and simulations are presented to illustrate the obtained results.

Predators catalyze an increase in chloroviruses by foraging on the symbiotic hosts of zoochlorellae

Virus population growth depends on contacts between viruses and their hosts. It is often unclear how sufficient contacts are made between viruses and their specific hosts to generate spikes in viral abundance. Here, we show that copepods, acting as predators, can bring aquatic viruses and their algal hosts into contact. Specifically, predation of the protist Paramecium bursaria by copepods resulted in a >100-fold increase in the number of chloroviruses in 1 d. Copepod predation can be seen as an ecological "catalyst" by increasing contacts between chloroviruses and their hosts, zoochlorellae (endosymbiotic algae that live within paramecia), thereby facilitating viral population growth. When feeding, copepods passed P. bursaria through their digestive tract only partially digested, releasing endosymbiotic algae that still supported viral reproduction and resulting in a virus population spike. A simple predator-prey model parameterized for copepods consuming protists generates cycle periods for viruses consistent with those observed in natural ponds. Food webs are replete with similar symbiotic organisms, and we suspect the predator catalyst mechanism is capable of generating blooms for other endosymbiont-targeting viruses.

A Simple Predator-Prey Population Model with Rich Dynamics

A non-smooth switched harvest on predators is introduced into a simple predator-prey model with logistical growth of the prey and a bilinear functional response. If the density of the predator is below a switched value, the harvesting rate is linear; otherwise, it is constant. The model links the well studied predator-prey model with constant harvesting to that with a proportional harvesting rate. It is shown that when the net reproductive number for the predator is greater than unity, the system is permanent and there may exist multiple positive equilibria due to the effects of the switched harvest, a saddle-node bifurcation, a limit cycle, and the coexistence of a stable equilibrium and a unstable circled inside limit cycle and a stable circled outside limit cycle. When the net reproductive number is less than unity, a backward bifurcation from a positive equilibrium occurs, which implies that the stable predator-extinct equilibrium may coexist with two coexistence equilibria. In this situation, reducing the net reproductive number to less than unity is not enough to enable the predator to go extinct. Numerical simulations are provided to illustrate the theoretical results. It seems that the model possesses new complex dynamics compared to the existing harvesting models.

Adaptive behaviour and multiple equilibrium states in a predator–prey model

There is evidence that multiple stable equilibrium states are possible in real-life ecological systems. Phenomenological mathematical models which exhibit such properties can be constructed rather straightforwardly. For instance, for a predator–prey system this result can be achieved through the use of non-monotonic functional response for the predator. However, while formal formulation of such a model is not a problem, the biological justification for such functional responses and models is usually inconclusive.In this note, we explore a conjecture that a multitude of equilibrium states can be caused by an adaptation of animal behaviour to changes of environmental conditions. In order to verify this hypothesis, we consider a simple predator–prey model, which is a straightforward extension of the classic Lotka–Volterra predator–prey model. In this model, we made an intuitively transparent assumption that the prey can change a mode of behaviour in response to the pressure of predation, choosing either “safe” of “risky” (or “business as usual”) behaviour. In order to avoid a situation where one of the modes gives an absolute advantage, we introduce the concept of the “cost of a policy” into the model. A simple conceptual two-dimensional predator–prey model, which is minimal with this property, and is not relying on odd functional responses, higher dimensionality or behaviour change for the predator, exhibits two stable co-existing equilibrium states with basins of attraction separated by a separatrix of a saddle point.

Timing and propagule size of invasion determine its success by a time-varying threshold of demographic regime shift.

Theory of invasion ecology indicates that the number of invading individuals (propagule size) and the timing of invasion are important for invasion success. Propagule size affects establishment success due to an Allee effect and the effect of demographic stochasticity, whereas the timing of invasion does so via niche opportunity produced by fluctuating predation pressure and resource abundance. We propose a synthesis of these two mechanisms by a time-varying dose-response curve where the dose is propagule size and the response is establishment probability. We show an example of the synthesis in a simple predator-prey model where successful invasion occurs as a demographic regime shift because of the bistability of the system. The two mechanisms are not independent, but simultaneously determine invasion success in our model. We found that positive growth rate of an invading species does not ensure its establishment, especially when its propagule size is small or when its growth rate is in a decreasing trend. We suggest the difficulty of understanding invasion process based on a dose-response curve of propagule size as no unique curve can be determined due to the effects of invasion timing (i.e., the threshold of demographic regime shift is time varied). The results of our model analysis also have an implication on the phase relationship between population cycles of predators and prey.

Assessing the utility of frequency dependent nudging for reducing biases in biogeochemical models

Bias errors, resulting from inaccurate boundary and forcing conditions, incorrect model parameterization, etc. are a common problem in environmental models including biogeochemical ocean models. While it is important to correct bias errors wherever possible, it is unlikely that any environmental model will ever be entirely free of such errors. Hence, methods for bias reduction are necessary. A widely used technique for online bias reduction is nudging, where simulated fields are continuously forced toward observations or a climatology. Nudging is robust and easy to implement, but suppresses high-frequency variability and introduces artificial phase shifts. As a solution to this problem Thompson et al. (2006) introduced frequency dependent nudging where nudging occurs only in prescribed frequency bands, typically centered on the mean and the annual cycle. They showed this method to be effective for eddy resolving ocean circulation models. Here we add a stability term to the previous form of frequency dependent nudging which makes the method more robust for non-linear biological models. Then we assess the utility of frequency dependent nudging for biological models by first applying the method to a simple predator–prey model and then to a 1D ocean biogeochemical model. In both cases we only nudge in two frequency bands centered on the mean and the annual cycle, and then assess how well the variability in higher frequency bands is recovered. We evaluate the effectiveness of frequency dependent nudging in comparison to conventional nudging and find significant improvements with the former.

Characterization of predator–prey dynamics, using the evolution of free energy in plasma turbulence

A simple dynamical cascade model for the evolution of free energy is considered in the context of gyrokinetic formalism. It is noted that the dynamics of free energy, that characterize plasma micro-turbulence in magnetic fusion devices, exhibit a predator–prey character. Various key features of predator–prey dynamics such as the time delay between turbulence and large-scale flow structures, or the intermittency of the dynamics are identified in the quasi-steady-state phase of the nonlinear gyrokinetic simulations. A novel prediction on the ratio of turbulence amplitudes in different parts of the wavenumber domain that follows from this simple predator–prey model is compared to a set of nonlinear simulation results and is observed to hold quite well in a large range of physical parameters. Detailed validation of the predator–prey hypothesis using nonlinear gyrokinetics provides a very important input for the effort to apprehend plasma micro-turbulence, since the predator–prey idea can be used as a very effective intuitive tool for understanding and designing efficient transport models.

Predator-Prey Modeling

Abstract. Predator-prey models are useful and often used in the environmental science field because they allow researchers to both observe the dynamics of animal populations and make predictions as to how they will develop over time. The objective of this project was to create five projections of animal populations based on a simple predator-prey model and explore the trends visible. Each case began with a set of initial conditions that produced different outcomes for the function of the population of rabbits and foxes over an 80 year time span. Using Euler's method, an Excel spreadsheet was developed to produce the values for the population of rabbits and foxes over the span of 80 years.

A Systematic Overview of Harvesting-Induced Maturation Evolution in Predator–Prey Systems with Three Different Life-History Tradeoffs

There are concerns that anthropogenic harvesting may cause phenotypic adaptive changes in exploited wild populations, in particular maturation at a smaller size and younger age. In this paper, we study the evolutionarily stable size at maturation of prey subjected to size-selective harvesting in a simple predator-prey model, taking into account three recognized life-history costs of early maturation, namely reduced fecundity, reduced growth, and increased mortality. Our analysis shows that harvesting large individuals favors maturation at smaller size compared to the unharvested system, independent of life-history tradeoff and the predator's prey-size preference. In general, however, the evolutionarily stable maturation size can either increase or decrease relative to the unharvested system, depending on the harvesting regime, the life-history tradeoff, and the predator's preferred size of prey. Furthermore, we examine how the predator population size changes in response to adaptive change in size at maturation of the prey. Surprisingly, in some situations, we find that the evolutionarily stable maturation size under harvesting is associated with an increased predator population size. This occurs, in particular, when early maturation trades off with growth rate. In total, we determine the evolutionarily stable size at maturation and associated predator population size for a total of forty-five different combinations of tradeoff, harvest regime, and predated size class.

Friend or foe: Conflicting demands and conditional risk taking by opportunistic scavengers

To optimize individual fitness, prey should balance predation threats with their own foraging demands. For prey that have additional competitive or positive interactions with their predators, however, little is known about the dynamics that regulate risk management behaviors. Our study focused on common marine gastropods (oyster drills) and one of their major predators: stone crabs. Notably, stone crabs not only prey on drills, but also serve as an ally by breeching the shells of eastern oysters. Subsequently, the foraging success of drills on oysters can be enhanced in the presence of crabs via scavenging on damaged oysters. We manipulated predator (crab) presence, resource (oyster) availability and prey (drill) hunger to explore how complex food-web interactions affect the risk management of predatory and scavenging gastropods. In mesocosm trials, we found that: (1) drills selected habitat closer to a central risk-reward patch as resource availability increased, although the presence of a stone crab generally stifled this movement toward the risk-reward patch; (2) in treatments without any live oysters provided as a foraging resource, drills were attracted by the presence of stone crabs despite the lack of an obvious incentive for this risk-enhancing behavior; (3) drills starved for 14days prior to entering mesocosms selected habitat without regard to predation threat, while drills that were not starved responded strongly to predator presence; and (4) drills became more aggregated as resource availability decreased, predation threat increased, or starvation level decreased. These results indicate that in resource-poor environments, drills demonstrate riskier behaviors than predicted by simple predator–prey models due to additional, positive interactions with their crab predators.

Dynamics of a delayed predator–prey model with predator migration

Based on the availability of prey and a simple predator–prey model, we propose a delayed predator–prey model with predator migration to describe biological control. We first study the existence and stability of equilibria. It turns out that backward bifurcation occurs with the migration rate as bifurcation parameter. The stability of the trivial equilibrium and the boundary equilibrium is delay-independent. However, the stability of the positive equilibrium may be delay-dependent. Moreover, delay can switch the stability of the positive equilibrium. When the positive equilibrium loses stability, Hopf bifurcation can occur. The direction and stability of Hopf bifurcation is derived by applying the center manifold method and the normal form theory. The main theoretical results are illustrated with numerical simulations.

Multiple extinction routes in stochastic population models

Isolated populations ultimately go extinct because of the intrinsic noise of elementary processes. In multipopulation systems extinction of a population may occur via more than one route. We investigate this generic situation in a simple predator-prey (or infected-susceptible) model. The predator and prey populations may coexist for a long time, but ultimately both go extinct. In the first extinction route the predators go extinct first, whereas the prey thrive for a long time and then also go extinct. In the second route the prey go extinct first, causing a rapid extinction of the predators. Assuming large subpopulation sizes in the coexistence state, we compare the probabilities of each of the two extinction routes and predict the most likely path of the subpopulations to extinction. We also suggest an effective three-state master equation for the probabilities to observe the coexistence state, the predator-free state, and the empty state.