The complex dynamics of two types of tritrophic food chain model systems when the species undergo spatial movements, modeling two real situations of marine ecosystem, are investigated in this study analytically and using numerical simulations. The study has been carried out with the objective to explore and compare the competitive effects of fish and molluscs species being the top predators, when phytoplankton and zooplankton species are undergoing spatial movements in the subsurface water. Reaction diffusion systems have been used to represent temporal evolution and spatial interaction among the species. The two model systems differ in an essential way that the top predators are generalist and specialist, respectively, in two models. “
The interest in the study of chaos in ecological systems has increased in the last couple of decades. Interactions of population in ecological systems are modeled by continuous time models and have been studied extensively in the literature [
Diffusion is a phenomenon by which the biological population spreads according to the irregular motion of each individual of the population. Diffusion equations have been used effectively to describe the movement of numerous animals in markrecapture studies [
In chemistry, seminal work of Turing [
The differential equation models for food chain were investigated by many researchers, and rich dynamical behaviors were found [
Two types of aquatic food chain model systems are studied in the presence of diffusion in the present paper. In our earlier work, the nondiffusive forms of the two nonlinear deterministic models of marine ecosystem were studied with a different aim of studying the effect of toxin on population dynamics [
The objective of this work is to investigate the effects of spatial interaction in marine ecosystem when top predator is either specialist or generalist in two different food chains. The investigation is carried out with the aim to understand the predation mechanism that will stabilize the dynamics of marine environment. This paper is organized as follows. In Section
The two nonlinear deterministic spatially extended preypredator model systems studied in this work represent two real situations of marine ecosystem, where the top predator is generalist predator in the first case and the top predator is specialist predator in the second case. In our earlier work [
The model system 1 has both kinds of predators, namely, specialist and generalist. The phytoplankton of density
where
The model system 2 describes the interaction of prey, specialist predator, and top specialist predator. At any location
In this section, we will perform a detailed study of the local dynamics of the two spatial model systems in a twodimensional domain. The conditions for the Turing instability to occur will be obtained by perturbing the homogeneous steadystate solution. The spatial model systems are studied under positive initial conditions and zero flux boundary conditions, given by
Before proceeding to the diffusiondriven analysis of the model system 1, it is worth discussing in brief the nonspatial model system. The biologically feasible equilibrium points of nonspatial form of model system (
For the linear stability analysis of spatial model (
where
The characteristic equation can be rewritten as
The interior equilibrium point
The spatially homogeneous state will be unstable provided that at least one eigenvalue of characteristic equation (
For (
The minimum of
The criterion for diffusive instability for the model system 1 is obtained at the critical wave number
First, we will discuss in brief the nonspatial model system. The biologically feasible equilibrium points of nonspatial form of model system 2 given in (
The linear stability analysis of spatial model (
The numerical simulations are carried out to understand the predation mechanism that will stabilize the dynamics of marine environment and to explore spatiotemporal pattern formation resulting from different types of predation. In this section, we investigate the effect of diffusion on the dynamics of the tritrophic food chain model systems. To explore the spatiotemporal dynamics of the model system, we have numerically solved the system of partial differential equations using finite difference method. Forward difference scheme is used for the reaction part. For diffusion part, central difference scheme is used for onedimensional case, and standard fivepoint explicit finite difference scheme is used for twodimensional diffusion terms. The simulation is carried out at different time levels for both onedimensional and twodimensional spatial model systems.
In onedimensional case model system 1 takes the form as given below:
The equations (
Simulation of model system (
At time
Chaotic attractor obtained for model system (
Figure
The twodimensional spatially extended model system 1 takes the form as given below:
The system of (
Two dimensional complex spatial patterns of prey, specialist, and generalist predators population of model (
Figure
The wave of chaos phenomenon, found in onedimensional space, is responsible for pattern formation seen in twodimensional spaces. This phenomenon has been observed in twospecies spatially extended model systems in the literature by different authors [
The model system 2 given in (
The model system (
Simulation of model system (
Chaotic attractor obtained for model system (
In the food chain ending with fish population as top predator, there has not been found any wave of chaos phenomenon in the onedimensional spatial domain. The spatial domain shows almost regular patterns over entire space at time
Although, in both of the food chains, one ending with mollusks and another ending with fish, the population density patterns in one dimension are chaotic (cf. Figures
In twodimensional case, the model system (
The system (
Two dimensional complex spatial patterns of prey, specialist, and top specialist predators population of model (
The spatiotemporal dynamics in the food chain model system which ends with specialist predator is much simpler than the dynamics of food chains ending with generalist predator. This study has investigated and found this interesting result and its possible cause. The numerical simulation shows that Wave of Chaos phenomenon, which is found only in food chain ending with generalist predator, is the possible cause of complex spatiotemporal patterns as seen in model system 1. The absence of WOC phenomenon in model system 2 leads to simple dynamics in spatial distribution of species.
After a detailed analytical and numerical simulation of model systems 1 and 2, in both one and two dimensions, we found that the spatially extended food chain model systems displayed chaotic dynamics leading to interesting spatial patterns. Also, from numerical simulation, we observed that patterns in a food chain with top predator as generalist predator are more complex than the patterns displayed by food chain ending with specialist predator.
The present study reveals that top specialist predators like fish can stabilize the dynamics of the marine ecosystem. This study suggests that presence of only the generalist predator as top predator in the ecosystem may destabilize the marine environment and can make it chaotic. This supports the study done by many ecologists that impact of more fishing will generally tend to reduce the abundance and diversity of parasites resulting in divergent and complex effects [
In the present paper, the pattern formation and spatiotemporal population dynamics in food chain model with diffusion are investigated. From the analytical and numerical results obtained, we found that diffusion plays an important role in the pattern formation of the tritrophic food chain model systems. Complex spatial patterns are obtained due to diffusion of the prey and predator population in a food web model. Further, we obtained chaotic dynamics in the simulation of onedimensional model system. It is observed that Wave of Chaos (WOC) mechanism is the possible cause for spatiotemporal pattern formation. The spatially extended model systems displayed chaotic attractor. The mathematical conditions for diffusiondriven instability are obtained. Numerical simulation of the two model systems studied in this paper shows that the food chain system with diffusion has rich dynamics including spatial patterns and chaotic dynamics.
We observed that a food chain model system with top predator as generalist as in model system 1 displays more complex spatial patterns (cf. Figure
We can conclude from the present study that the presence of fish in marine ecosystem makes the food chain model more stable and helps in making marine environment ecofriendly and nonchaotic whereas effects of overfishing can be divergent and complex. This study supports the ecological study that overfishing can lead to global decline in fish parasites which are critical part of biodiversity, and their loss could substantially impact ecosystem function This study will have a direct impact on fisheries policy [
This study reveals that a large variety of different patterns and spatiotemporal dynamics can be found in a food chain system with spatial interaction modeled by selfdiffusion. Further research is required to explore the dynamical complexity of spatially extended food chain models with more numbers of species.
Description and numerical values of parameters of model 1 and model 2.
Model  Parameters  Numerical values  Figure  Reference 


1.93, 0.06, 1, 10, 0.189, 0.5 
Figures 
Letellier and AzizAlaoui [  
Model 1 

10, 0.405, 10, 0.03, 1, 20  Upadhyay et al. [  

1, 1, 1  Current study  
 

1.75, 0.05, 1, 10, 0.2, 0.8  [  
Model 2 

10, 1.45, 10, 0.1, 1, 20  Figures 
[ 

1, 1, 1  Current study 
The present research of the author is supported by DST under IUATCPhase 2, Project No. SR/RCUKDST/IUATC Phase 2/2012IITM (G) and IIT Mandi under the Project No. IITM/SG/NTK/008. The author would like to thank the referees for their constructive comments, which led to a significant improvement of the original paper.