Discrete-time Host-parasitoid Model Research Articles

The interplay between multiple control mechanisms in a host–parasitoid system: a discrete-time stage-structured modelling approach

We propose a discrete-time host–parasitoid model with stage structure in both species. For this model, we establish conditions for the existence and global stability of the extinction and parasitoid-free equilibria as well as conditions for the existence and local stability of an interior equilibrium and system persistence. We study the model numerically to examine how pesticide spraying may interact with natural enemies (parasitoids) to control the pest (host) species. We then extend the model to an impulsive difference system that incorporates both periodic pesticide spraying and augmentation of the natural enemies to suppress the pest population. For this system, we determine when the pest-eradication periodic solution is globally attracting. We also examine how varying the control measures (pesticide concentration, natural enemy augmentation and the frequency of applications) may lead to different pest outbreak or persistence outcomes when eradication does not occur.

A Host-parasitoid Dynamics with Allee and Refuge effects

We examine a discrete-time host–parasitoid model incorporating simultaneously an Allee and a refuge effect on the host. We investigate existence of a positive fixed point, local asymptotic stability, global stability of the fixed points and bifurcations. Numerical examples are given for verification of the theoretical results and we compare the model with existing data from the literature.

Stability and bifurcation analysis of a discrete-time host-parasitoid model with Holling III functional response

<abstract><p>We study the dynamical properties of a discrete-time host-parasitoid model with Holling type III functional response. It is shown that flip bifurcation and Neimark-Sacker bifurcation occur in certain parameter regimes. A sufficient condition based on the model parameters for which both populations can coexist is derived. The boundedness, existence and local stability of the unique equilibrium are proved. In addition, the numerical simulations have been done, in addition to supporting the analytical findings, more behaviors are extracted from the model in a two-dimensional parameter space. Finally, we emphasize the importance of clearly presenting biological assumptions that are inherent to the structure of a discrete model.</p></abstract>

Local Stability Analysis and Bifurcations of a Discrete-Time Host-Parasitoid Model

In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynamics of this system. We conduct the bifurcation analysis with respect to intrinsic growth rate r and searching efficiency a. Many forms of complex dynamics such as chaos, periodic windows are observed. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for b≠a where a,b are searching efficiency. We study stable and unstable manifolds for different equilibrium points and coexistence of different attractors for this non-dimensionalize system. Without the parasitoid, the host population follows the dynamics of the Ricker model.

Neimark-Sacker Bifurcation and Chaotic Behaviour of a Modified Host–Parasitoid Model

Abstract We study some qualitative behaviour of a modified discrete-time host–parasitoid model. Modification of classical Nicholson–Bailey model is considered by introducing Pennycuick growth function for the host population. Furthermore, the existence and uniqueness of positive equilibrium point of proposed system is investigated. We prove that the positive solutions of modified system are uniformly bounded and the unique positive equilibrium point is locally asymptotically stable under certain parametric conditions. Moreover, it is also investigated that system undergoes Neimark–Sacker bifurcation by using standard mathematical techniques of bifurcation theory. Complexity and chaotic behaviour are confirmed through the plots of maximum Lyapunov exponents. In order to stabilise the unstable steady state, the feedback control strategy is introduced. Finally, in order to support theoretical discussions, numerical simulations are provided.

Global stability and Neimark-Sacker bifurcation of a host-parasitoid model

ABSTRACTWe investigate qualitative behaviour of a density-dependent discrete-time host-parasitoid model. Particularly, we study boundedness of solutions, existence and uniqueness of positive steady-state, local and global asymptotic stability of the unique positive equilibrium point and rate of convergence of modified host-parasitoid model. Moreover, it is also proved that the system undergoes Neimark-Sacker bifurcation with the help of bifurcation theory. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long-term behaviour over a broad range of parameters. The computation of the maximum Lyapunov exponents confirm the presence of chaotic behaviour in the model.

Allee Effect and Stability in a Discrete-Time Host-Parasitoid Model

Allee Effect and Stability in a Discrete-Time Host-Parasitoid Model

A discrete-time host–parasitoid model with an Allee effect

We introduce a discrete-time host–parasitoid model with a strong Allee effect on the host. We adapt the Nicholson–Bailey model to have a positive density dependent factor due to the presence of an Allee effect, and a negative density dependence factor due to intraspecific competition. It is shown that there are two scenarios, the first with no interior fixed points and the second with one interior fixed point. In the first scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and an exclusion region in which the host survives and tends to its carrying capacity. In the second scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and a coexistence region where both species survive.

Host-Parasitoid Model With Intraspecific Competitions

In this paper a discrete-time host-parasitoid model with intraspecific competitions is proposed. Phase portraits are drawn for different types of intraspecific competitions to notice the effects of intraspecific competitions for biologically reasonable range of parameter values. Bifurcation analysis is done with respect to instantaneous search rate as well as handling time for different types of intraspecific competitions. Many forms of complex dynamics such as chaos, periodic windows etc. are observed. The stable coexistence as well as oscillatory coexistence of host and parasitoid are shown under different types of intraspecific competitions. The Hopf point and attractor crises exist for different intraspecific competitions.

Discrete-time host–parasitoid models with pest control

We propose a simple discrete-time host–parasitoid model to investigate the impact of external input of parasitoids upon the host–parasitoid interactions. It is proved that the input of the external parasitoids can eventually eliminate the host population if it is above a threshold and it also decreases the host population level in the unique interior equilibrium. It can simplify the host–parasitoid dynamics when the host population practices contest competition. We then consider a corresponding optimal control problem over a finite time period. We also derive an optimal control model using a chemical as a control for the hosts. Applying the forward–backward sweep method, we solve the optimal control problems numerically and compare the optimal host populations with the host populations when no control is applied. Our study concludes that applying a chemical to eliminate the hosts directly may be a more effective control strategy than using the parasitoids to indirectly suppress the hosts.

The impact of parasitoid emergence time on host–parasitoid population dynamics

We investigate the effect of parasitoid phenology on host–parasitoid population cycles. Recent experimental research has shown that parasitized hosts can continue to interact with their unparasitized counterparts through competition. Parasitoid phenology, in particular the timing of emergence from the host, determines the duration of this competition. We construct a discrete-time host–parasitoid model in which within-generation dynamics associated with parasitoid timing is explicitly incorporated. We found that late-emerging parasitoids induce less severe, but more frequent, host outbreaks, independent of the choice of competition model. The competition experienced by the parasitized host reduces the parasitoids’ numerical response to changes in host numbers, preventing the ‘boom-bust’ dynamics associated with more efficient parasitoids. We tested our findings against experimental data for the forest tent caterpillar ( Malacosoma disstria Hübner) system, where a large number of consecutive years at a high host density is synonymous with severe forest damage.

Dynamics of a host–parasitoid model with Allee effect for the host and parasitoid aggregation

In this paper, a discrete-time host–parasitoid model is investigated. Two biological phenomena, the Allee effect of the host population and the aggregation of the parasitism, are considered in our mathematical model. Through extensive numerical simulations, we gain some interesting findings related to Allee effect from this research. Firstly, the ranges of parameter, in which the population dynamics is chaos, are compressed when Allee effect is added. Secondly, the sensitivity to initial conditions of the host–parasitoid system decreased after adding Allee effect. Thirdly, without Allee effect, we observed two complicated dynamics, intermittent chaos and supertransients. However, when Allee effect is included, these two phenomena are replaced by another kind of phenomenon-period alternation, where chaos is eliminated. From above three novel findings, it can be concluded that dynamic complexities are alleviated by Allee effect. This conclusion is crucial in resolving the discrepancy between real population dynamics and theoretical predictions. Furthermore, the importance of this research is to help us understand the mechanisms inducing the irregular fluctuations of the natural populations.

Allee effects in a discrete-time host-parasitoid model with stage structure in the host

We study a single-species population model with two stages,adults and juveniles, and the model with Allee effects. In thesemodels, the fertility rate of an adult individual is assumed to bedensity dependent on the total adult population size and thetransition probability from juvenile to adult over each time unitis assumed to be a constant. Both models exhibit a boundary$2$-cycle. Population persistence can occur for the model withoutthe Allee effects. However, there exists a population thresholdbelow which the population will go to extinction if the Alleeeffects are considered. We also propose a host-parasitoid modelwith stage structure in the host. Both populations can coexistwith each other under some conditions if Allee effects areignored. On the other hand, there exists a host populationthreshold below which both populations become extinct if Alleeeffects are incorporated into the interaction.

Host-parasitoid dynamics of a generalized Thompson model

A discrete-time host-parasitoid model including host-density dependence and a generalized Thompson escape function is analyzed. This model assumes that parasitoids are egg-limited but not search-limited, and is proven to exhibit five types of dynamics: host failure in which the host goes extinct in the parasitoid's presence or absence, unconditional parasitoid failure in which the parasitoid always goes extinct while the host persists, conditional parasitoid failure in the host and the parasitoid go extinct or coexist depending on the initial host-parasitoid ratio, parasitoid driven extinction in which the parasitoid invariably drives the host to extinction, and coexistence in which the host and parasitoid coexist about a global attractor. The latter two dynamics only occur when the parasitoid's maximal rate of growth exceeds the host's maximal rate of growth. Moreover, coexistence requires parasitism events to be sufficiently aggregated. Small additive noise is proven to alter the dynamical outcomes in two ways. The addition of noise to parasitoid driven extinction results in random outbreaks of the host and parasitoid with varying intensity. Additive noise converts conditional parasitoid failure to unconditional parasitoid failure. Implications for classical biological control are discussed.

Allee effects in a discrete-time host-parasitoid model

A discrete-time Beverton-Holt stock recruitment model with Allee effects is proposed and studied. Its asymptotic dynamics are then used to investigate a host-parasitoid model with Allee effects occurring in the host population. There exists a population threshold of which initial host population abundance must exceed so that the host population may persist. Both populations are more likely to become extinct when Allee effects are incorporated into this host-parasitoid interaction.