Modeling and analysis of a predator-prey model with disease in the prey

A system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. Mathematical analyses of the model equations with regard to invariance of non-negativity, boundedness of solutions, nature of equilibria, permanence and global stability are analyzed. If the coefficient in conversing prey into predator k=k(0) is constant (independent of delay tau;, gestation period), we show that positive equilibrium is locally asymptotically stable when time delay tau; is suitable small, while a loss of stability by a Hopf bifurcation can occur as the delay increases. If k=k(0)e(-dtau;) (d is the death rate of predator), numerical simulation suggests that time delay has both destabilizing and stabilizing effects, that is, positive equilibrium, if it exists, will become stable again for large time delay. A concluding discussion is then presented.     

Chaotic dynamics of a three species prey-predator competition model with bionomic harvesting due to delayed environmental noise as external driving force

We consider a biological economic model based on prey-predator interactions to study the dynamical behaviour of a fishery resource system consisting of one prey and two predators surviving on the same prey. The mathematical model is a set of first order non-linear differential equations in three variables with the population densities of one prey and the two predators. All the possible equilibrium points of the model are identified, where the local and global stabilities are investigated. Biological and bionomical equilibriums of the system are also derived. We have analysed the population intensities of fluctuations i.e., variances around the positive equilibrium due to noise with incorporation of a constant delay leading to chaos, and lastly have investigated the stability and chaotic phenomena with a computer simulation.     

Time delay can enhance spatio-temporal chaos in a prey–predator model

In this paper we explore how the time delay induced Hopf-bifurcation interacts with Turing instability to determine the resulting spatial patterns. For this study, we consider a delayed prey–predator model with Holling type-II functional response and intra-specific competition among the predators. Analytical criteria for the delay induced Hopf-bifurcation and for the delayed spatio-temporal model are provided with numerical example to validate the analytical results. Exhaustive numerical simulation reveals the appearance of three types of stationary patterns, cold spot, labyrinthine, mixture of stripe-spot and two non-stationary patterns, quasi-periodic and spatio-temporal chaotic patterns. The qualitative features of the patterns for the non-delayed and the delayed spatio-temporal model are the same but their occurrence is solely controlled by the temporal parameters, rate of diffusivity and magnitude of the time delay. It is evident that the magnitude of time delay parameter beyond the Hopf-bifurcation threshold mostly produces spatio-temporal chaotic patterns.     

Modeling the Fear Effect in Predator–Prey Interactions with Adaptive Avoidance of Predators

Recent field experiments on vertebrates showed that the mere presence of a predator would cause a dramatic change of prey demography. Fear of predators increases the survival probability of prey, but leads to a cost of prey reproduction. Based on the experimental findings, we propose a predator–prey model with the cost of fear and adaptive avoidance of predators. Mathematical analyses show that the fear effect can interplay with maturation delay between juvenile prey and adult prey in determining the long-term population dynamics. A positive equilibrium may lose stability with an intermediate value of delay and regain stability if the delay is large. Numerical simulations show that both strong adaptation of adult prey and the large cost of fear have destabilizing effect while large population of predators has a stabilizing effect on the predator–prey interactions. Numerical simulations also imply that adult prey demonstrates stronger anti-predator behaviors if the population of predators is larger and shows weaker anti-predator behaviors if the cost of fear is larger.     

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.     

一类考虑收获的时滞捕食系统的稳定性与Hopf分支研究

给出了一类考虑收获的时滞捕食系统的局部稳定性判断,并由规范型理论和中心流形定理推导出了Hopf分支的方向、稳定性等条件,最后给出了两个数值模拟例子验证了结论的正确性.     

Delays in recruitment at different trophic levels: Effects on stability

Predator prey models in which there is a delay in recruitment in both species and only adults interact are formulated and studied. Stability results show that the length of delays in recruitment in the prey are more critical than the length of recruitment delays in the predator. Thus the destabilizing effect of recruitment delays may be less important in higher trophic levels than some single species models indicate. Variance in the recruitment delay is shown to be an important stabilizing influence.     

The oddity effect drives prey choice but not necessarily attack time

The tendency of predators to preferentially attack phenotypically odd prey in groups (the oddity effect) is a clear example of how predator cognition can impact behaviour and morphology in prey. Through targeting phenotypically odd prey, predators are thought to avoid the cognitive constraints that delay and limit the success of attacks on homogenous prey groups (the confusion effect). In addition to influencing which prey a predator will attack, the confusion and oddity effects would also predict that attacks on odd prey can occur more rapidly than attacking the majority prey type, as odd prey are more easily targeted, but this prediction has yet to be tested. Here, we used kerri tetra fish, Inpaichthys kerri, presented with mixed phenotypic groups of Daphnia dyed red or black to investigate whether odd prey in groups are preferentially attacked and whether these attacks were faster than those on the majority prey type. In agreement with previous work, odd prey were targeted and attacked more often than expected from their frequency in the prey groups, regardless of whether the odd prey was red in a group of black prey or vice versa. However, no difference was found in the time taken to attack odd vs. majority prey items, contrary to our predictions. Our results suggest that the time taken to make an attack is determined by a wider range of factors or is subject to greater variance than the choice of which prey is selectively targeted in a group.     

Prey availability and eclosion-help of callow workers in the formicine ant Camponotus mus

Summary In the ant Camponotus mus the eclosion of imagines from the cocoons was shown to be controlled by nurse workers, which broke the cocoons and assisted the callows to energe. Short-term colony prey deprivation modified the eclosion-help behavior resulting in a delay in cocoon opening times. During long-term prey deprivation, no new cocoons were spun; the starved larvae grew and pupated when prey was again available. These findings as a colony strategy adapted to the changing condition of prey availabitity in a temperate habitat are discussed.     

The effect of long time delays in predator-prey systems

Past studies have indicated that a time delay longer than the natural period of a system will generally cause instability; however here it is shown that including long maturational time delays in a general predator-prey model need not have this effect. In each of the three cases studied (a predator delay, a prey delay, and both), local stability can persevere despite the presence of arbitrarily long time delays. This perseverence depends upon an interaction between delayed and undelayed features of the model. Delayed processes always act to destabilize the model. For example, prey self-regulation, usually a source of stability, becomes destabilizing if subject to a long delay. However, the effect of such a delay is offset by undelayed regulatory processes, such as a stabilizing functional reponse. In addition, the adverse effects of delayed predator recruitment can be reduced by the nonreproductive component of the numerical reponse, a feature not usually involved in determining stability. Finally, it is shown that long time delays are not necessarily more disruptive than short delays; it cannot be assumed that lengthening a time delay progessively reduces stability.     

Modelling hiding behaviour in a predator-prey system by both integer order and fractional order derivatives

Most of the preys are well aware of sensing predation risk. Consequently, to escape from predators they usually adopt several defense mechanisms, specially refuge themselves to become invulnerable. In view of this, a mathematical model has been formulated incorporating prey refuge, where it is assumed that prey refuge is a function of predators availability in the system. It is shown that the model system is well-posed. It has been found that the hiding level and consumption rate of predators have a suitable interrelation between them. Both the parameters act as Hopf bifurcation parameters, but they play opposite role in case of stabilization of the system dynamics. Also, hiding level plays crucial role in maintaining the mean density of both the populations. Furthermore, as hiding behaviour of prey is not instantaneous, so a time delay, namely hiding delay has been introduced to make the model system more realistic and it is observed that the delay parameter destabilizes the system. Modelling approach through fractional calculus has been further deployed to study how the process of forgetting life history influences the dynamical intricacy of the population level dynamics. All the analytical findings have been testified by proper numerical performances.     

Warning signals and predator-prey coevolution

Theories of the evolution of warning signals are typically expressed using analytic and computational models, most of which attribute aspects of predator psychology as the key factors facilitating the evolution of warning signals. Sherratt provides a novel and promising perspective with a model that considers the coevolution of predator and prey populations, showing how predators may develop a bias towards attacking cryptic prey in preference to conspicuous prey. Here, we replicate the model as an individual-based simulation and find, in accordance with Sherratt, that predators evolve a bias towards attacking cryptic prey. We then use a Monte Carlo simulation to calculate the relative survivorships of cryptic and conspicuous prey and stress that, as it stands, the model does not predict the evolution or stability of warning signals. We extend the model by giving predators continuous attack strategies and by allowing the evolution of prey conspicuousness: results are robust to the first modification but, in all cases, cryptic prey always enjoy a higher survivorship than conspicuous prey. When conspicuousness is allowed to evolve, prey quickly evolve towards crypsis, even when runaway coevolution is enabled. Sherratt's approach is promising, but other aspects of predator psychology, besides their innate response, remain vital to our understanding of warning signals.     

Hopf bifurcation in three-species food chain models with group defense.

Three-species food-chain models, in which the prey population exhibits group defense, are considered. Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model without delay undergoes a sequence of Hopf bifurcations. In the model with delay it is shown that using a delay as a bifurcation parameter, a Hopf bifurcation can also occur in this case. These occurrences may be interpreted as showing that a region of local stability (survival) may exist even though the positive steady states are unstable. A computer code BIFDD is used to determine the stability of the bifurcation solutions of a delay model.     

The profit of senescence in phytoseiid mites

Summary A long post-reproductive period in females under optimal conditions is common among phytoseiid mites which are specialized predators of Tetranychus-species. The resulting long life span provides the possibility to postpone or delay egg production during periods of prey scarcity, theoretically in both virgin and fertilized females. The latter are better as colonists. These specialized predators are also the most prolific phytoseiids. The unusual combination of large reproductive capacity and long life might be explained as an adaptation to widely fluctuating prey densities.     

Incorporating prey refuge in a prey–predator model with a Holling type I functional response: random dynamics and population outbreaks

A prey–predator discrete-time model with a Holling type I functional response is investigated by incorporating a prey refuge. It is shown that a refuge does not always stabilize prey–predator interactions. A prey refuge in some cases produces even more chaotic, random-like dynamics than without a refuge and prey population outbreaks appear. Stability analysis was performed in order to investigate the local stability of fixed points as well as the several local bifurcations they undergo. Numerical simulations such as parametric basins of attraction, bifurcation diagrams, phase plots and largest Lyapunov exponent diagrams are executed in order to illustrate the complex dynamical behavior of the system.     

The advantage of alternative tactics of prey and predators depends on the spatial pattern of prey and social interactions among predators

Individual variation in behavioral strategies is ubiquitous in nature. Yet, explaining how this variation is being maintained remains a challenging task. We use a spatially-explicit individual-based simulation model to evaluate the extent to which the efficiency of an alternative spacing tactic of prey and an alternative search tactic of predators are influenced by the spatial pattern of prey, social interactions among predators (i.e., interference and information sharing) and predator density. In response to predation risk, prey individuals can either spread out or aggregate. We demonstrate that if prey is extremely clumped, spreading out may help when predators share information regarding prey locations and when predators shift to area-restricted search following an encounter with prey. However, dispersion is counter-selected when predators interact by interference, especially under high predator density. When predators search for more randomly distributed prey, interference and information sharing similarly affect the relative advantage of spreading out. Under a clumped prey spatial pattern, predators benefit from shifting their search tactic to an area-restricted search following an encounter with prey. This advantage is moderated as predator density increases and when predators interact either by interference or information sharing. Under a more random prey pattern, information sharing may deteriorate the inferior search tactic even more, compared to interference or no interaction among predators. Our simulation clarifies how interactions among searching predators may affect aggregation behavior of prey, the relative success of alternative search tactics and their potential to invade established populations using some other search or spacing tactics.     

Experiments and a model examining learning in the area-restricted search behavior of ferrets (Mustela putorius furo)

Area-restricted searches have been described as important componentsof the foraging behavior of many organisms. It is unclear, however,whether individual foragers can use learning to fine-tune theirsearches, or even whether these searches are efficiently performed.I used a simulation model to make qualitative predictions aboutsearch behavior in a laboratory system. The simulation modelindicates that the sinuosity and path length of searches stronglyaffect search efficiency. The model predicts that, for a rate-maximizingforager, path length should increase and search sinuosity shoulddecrease as prey become less clumped. Foraging animals may thereforebe selected to learn the path length and sinuosity of searchesin response to changing degrees of dumping of prey. These predictionswere tested in a laboratory system involving ferrets (Mustelaputorius furo) foraging for oil-drop "prey items." Search pathschanged in a graded manner to experimental manipulations ofthe dumping of prey. As predicted by the model, ferrets learnedto perform longer and less sinuous search paths as prey becameless clumped. This study provides the first evidence that area-restrictedsearch behavior is learned and can be fine-tuned to efficientlyexploit different spatial distributions of food.     

草间小黑蛛对棉铃虫幼虫的捕食作用及其模拟模型的研究——Ⅱ.捕食者——多种猎物系统的研究

草间小黑蛛Erigonidium graminicolum(Sundevall)与单种猎物的作用关系已在上文述及(李超等,1982),然而草间小黑蛛在自然情形下是一种多食性的捕食性天敌,它的捕食对象不止棉铃虫Heliothis armigera H(?)bner一种。我们在对草间小黑蛛捕食行为的观察中也发现,捕食对象除棉铃虫外还有棉蚜等,并且在棉花上主要是徘徊式地捕获猎物。这样,在草间小黑蛛-棉铃虫作用系统的动态中就必须考虑替代食物的影响。替代食物的存在使捕食者对不同种类的猎物由于有不同的相对喜好程度而产生了对食物的选择,而该相对喜好程度的     

The efficiency of adaptive search tactics for different prey distribution patterns: a simulation model based on the behaviour of juvenile plaice

Juvenile plaice Pleuronectes platessa are particularly useful for studying forager search behaviour because their search paths are essentially two dimensional, and punctuated by natural stops. Their prey occur in a range of natural distributions from highly aggregated to over‐dispersed. Juvenile plaice use area‐restricted search near aggregated prey and extensive search, consisting of longer moves and fewer turns, between aggregations and when searching for dispersed prey. They search for less conspicuous prey items mainly in the pauses between movements. This saltatory search behaviour contrasts with the continuous search that is usually assumed in search models. A simulation model of saltatory search behaviour showed that a strategy combining extensive and intensive search allows the efficient exploitation of a range of natural prey distribution patterns, and that it is particularly effective when the search behaviour is controlled by perceived prey density. This allows the predator to respond to the localized aggregations which often occur in nature. The selective use of intensive search was more efficient than the continuous use of extensive search even in prey distribution patterns that were statistically over‐dispersed.     

Qualitative and Quantitative Analyses of the Echolocation Strategies of Bats on the Basis of Mathematical Modelling and Laboratory Experiments

Prey pursuit by an echolocating bat was studied theoretically and experimentally. First, a mathematical model was proposed to describe the flight dynamics of a bat and a single prey. In this model, the flight angle of the bat was affected by angles related to the flight path of the single moving prey, that is, the angle from the bat to the prey and the flight angle of the prey. Numerical simulation showed that the success rate of prey capture was high, when the bat mainly used the angle to the prey to minimize the distance to the prey, and also used the flight angle of the prey to minimize the difference in flight directions of itself and the prey. Second, parameters in the model were estimated according to experimental data obtained from video recordings taken while a Japanese horseshoe bat (Rhinolphus derrumequinum nippon) pursued a moving moth (Goniocraspidum pryeri) in a flight chamber. One of the estimated parameter values, which represents the ratio in the use of the angles, was consistent with the optimal value of the numerical simulation. This agreement between the numerical simulation and parameter estimation suggests that a bat chooses an effective flight path for successful prey capture by using the angles. Finally, the mathematical model was extended to include a bat and prey. Parameter estimation of the extended model based on laboratory experiments revealed the existence of bat’s dynamical attention towards prey, that is, simultaneous pursuit of prey and selective pursuit of respective prey. Thus, our mathematical model contributes not only to quantitative analysis of effective foraging, but also to qualitative evaluation of a bat’s dynamical flight strategy during multiple prey pursuit.