Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications (original) (raw)
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We present a spatial host-parasitoid model where individuals move on a square lattice of patches. Local interactions between hosts and parasitoids within patches are described by the Nicholson-Bailey model. Dispersal between patches is represented by a series of movement events from a patch to neighbouring patches. We study the effect of the number of movement events on the stability of the host-parasitoid system. The aim of this work is to determine conditions on this number for using a reduced model (called aggregated model) to predict the total host and parasitoid population dynamics. When the number of movement events is small, the system is usually persistent and spatial patterns are observed, such as spiral waves or chaotic dynamics. We show that when this number is larger than a critical value, spatial homogeneity is observed after some transient dynamics and the system does not persist; in that case the reduced model can be used. Our results show that the critical value is relatively small and that the reduced model can be used in realistic situations.
Patterns of 2-Year Population Cycles in Spatially Extended Host–Parasitoid Systems
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Coexisting but temporally separated cohorts of insects with a multiannual life cycle may have dissimilar average abundance, resulting in periodically fluctuating population density. In the case of the boreal moth genus Xestia with a 2-year life cycle and a distinct abundance difference between the two coexisting cohorts, empirical results and a simple model suggest that the oscillatory dynamics are maintained by interaction with a parasitoid wasp. Here we report theoretical results on a spatially extended version of the basic model and relate the modeling results to empirical observations. A spatially extended model may have domains oscillating in different phases as is the case between western and eastern Finnish Lapland. Spatial heterogeneity tends to fix the location of phase boundaries. In contrast, spatially homogeneous temporal fluctuations tend to synchronize populations in large regions. ]
Global Behavior and Bifurcation in a Class of Host–Parasitoid Models with a Constant Host Refuge
Qualitative Theory of Dynamical Systems, 2020
In this paper, by using the analytical approach, we investigate the global behavior and bifurcation in a class of host-parasitoid models when a constant number of the hosts are safe from parasitism. We find the conditions for the existence and stability of the equilibria. We detect the existence of the Neimark-Sacker bifurcation under certain conditions. We explicitly derived the approximation of the limit curve depending on the parameters that appear in the model. We show that a locally asymptotically stable equilibrium can never be transformed into unstable by increasing a constant number of hosts that are using a refuge. Specially, we consider the effect of constant host refuge in (S), (HV), and (PP) models.The obtained results show that the constant number of hosts in refuge affects the qualitative behavior of these models in comparison to the same models without refuge. The theory is confirmed and illustrated numerically.
Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems
Journal of mathematical biology, 2007
When searching for hosts, parasitoids are observed to aggregate in response to chemical signalling cues emitted by plants during host feeding. In this paper we model aggregative parasitoid behaviour in a multi-species host-parasitoid community using a system of reaction-diffusion-chemotaxis equations. The stability properties of the steady-states of the model system are studied using linear stability analysis which highlights the possibility of interesting dynamical behaviour when the chemotactic response is above a certain threshold. We observe quasi-chaotic dynamic heterogeneous spatio-temporal patterns, quasi-stationary heterogeneous patterns and a destabilisation of the steady-states of the system. The generation of heterogeneous spatio-temporal patterns and destabilisation of the steady state are due to parasitoid chemotactic response to hosts. The dynamical behaviour of our system has both mathematical and ecological implications and the concepts of chemotaxis-driven instability and coexistence and ecological change are discussed.
In this paper, we present and analyze a spatio-temporal eco-epidemiological model of a prey predator system where prey population is infected with a disease. The prey population is divided into two categories, susceptible and infected. The susceptible prey is assumed to grow logistically in the absence of disease and predation. The predator population follows the modified Leslie-Gower dynamics and predates both the susceptible and infected prey population with Beddington-DeAngelis and Holling type II functional responses, respectively. The boundedness of solutions, existence and stability conditions of the biologically feasible equilibrium points of the system both in the absence and presence of diffusion are discussed. It is found that the disease can be eradicated if the rate of transmission of the disease is less than the death rate of the infected prey. The system undergoes a transcritical and pitchfork bifurcation at the Disease Free Equilibrium Point when the prey infection rate crosses a certain threshold value. Hopf bifurcation analysis is also carried out in the absence of diffusion, which shows the existence of periodic solution of the system around the Disease Free Equilibrium Point and the Endemic Equilibrium Point when the ratio of the rate of intrinsic growth rate of predator to prey crosses a certain threshold value. The system remains locally asymptotically stable in the presence of diffusion around the disease free equilibrium point once it is locally asymptotically stable in the absence of diffusion. The Analytical results show that the effect of diffusion can be managed by appropriately choosing conditions on the parameters of the local interaction of the system. Numerical simulations are carried out to validate our analytical findings. (D. Melese).
Stability of a certain class of a host–parasitoid models with a spatial refuge effect
Journal of Biological Dynamics, 2019
A certain class of a host-parasitoid models, where some host are completely free from parasitism within a spatial refuge is studied. In this paper, we assume that a constant portion of host population may find a refuge and be safe from attack by parasitoids. We investigate the effect of the presence of refuge on the local stability and bifurcation of models. We give the reduction to the normal form and computation of the coefficients of the Neimark-Sacker bifurcation and the asymptotic approximation of the invariant curve. Then we apply theory to the three well-known host-parasitoid models, but now with refuge effect. In one of these models Chenciner bifurcation occurs. By using package Mathematica, we plot bifurcation diagrams, trajectories and the regions of stability and instability for each of these models.
Host-parasitoid spatial dynamics in heterogeneous landscapes
Oikos, 2007
This paper explores the effect of spatial processes in a heterogeneous environment on the dynamics of a hostparasitoid interaction. The environment consists of a lattice of favourable (habitat) and hostile (matrix) hexagonal cells, whose spatial distribution is measured by habitat proportion and spatial autocorrelation (inverse of fragmentation). At each time step, a fixed fraction of both populations disperses to the adjacent cells where it reproduces following the Nicholson-Bailey model. Aspects of the dynamics analysed include extinction, stability, cycle period and amplitude, and the spatial patterns emerging from the dynamics.
Chaos in functional response host–parasitoid ecosystem models
Chaos, Solitons & Fractals, 2002
Natural populations whose generations are non-overlapping can be modeled by dierence equations that describe how the population evolves in discrete time-steps. In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, the investigations of complex population dynamics have mainly concentrated on single populations and no higherdimensional ecological systems. In this study, in order to simulate cyclic eects due to changes in parasitoid's behavior, Holling type II and III functional response functions are applied to host±parasitoid models, respectively. For each model, the complexities include (1) chaotic bands with periodic windows, pitchfork and tangent bifurcations, and attractor crises, (2) non-unique dynamics, meaning that several attractors coexist, (3) intermittency, and (4) supertransients. Ó : S 0 9 6 0 -0 7 7 9 ( 0 1 ) 0 0 0 6 3 -7
A Density-Dependent Host-parasitoid Model with Stability, Bifurcation and Chaos Control
Mathematics
The aim of this article is to study the qualitative behavior of a host-parasitoid system with a Beverton–Holt growth function for a host population and Hassell–Varley framework. Furthermore, the existence and uniqueness of a positive fixed point, permanence of solutions, local asymptotic stability of a positive fixed point and its global stability are investigated. On the other hand, it is demonstrated that the model endures Hopf bifurcation about its positive steady-state when the growth rate of the consumer is selected as a bifurcation parameter. Bifurcating and chaotic behaviors are controlled through the implementation of chaos control strategies. In the end, all mathematical discussion, especially Hopf bifurcation, methods related to the control of chaos and global asymptotic stability for a positive steady-state, is supported with suitable numerical simulations.
A Host-parasitoid Dynamics with Allee and Refuge effects
European Journal of Pure and Applied Mathematics
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.