Ranjit Kumar Upadhyay

Complex Population Dynamics in Heterogeneous Environments: Effects of Random and Directed Animal Movements

Abstract In this paper, we have investigated the complex dynamics of a one-dimensional spatial nonlinear coupled reaction-diffusion system with a Holling type IV functional response, akin to standard Michaelis-Menten inhibitory kinetics. Prey-taxis is included in a general reaction-diffusion equation to incorporate the active movement of predator species towards regions with high prey concentrations or if the predator is following some sort of cue (such as odor) to find the prey. We have carried out stability analysis of both the non-spatial model without diffusive spreading and of the spatial model. We performed extensive computer simulations to identify various parameter ranges for stable homogeneous solution. Our findings specifically elucidate the role of predator diffusion and prey-taxis in controlling emergent structures, and transitions towards spatio-temporal chaos. We observe that the increasing predator random movement and moderate value of prey-taxis stabilize the system.

DIFFUSION-DRIVEN INSTABILITIES AND SPATIO-TEMPORAL PATTERNS IN AN AQUATIC PREDATOR–PREY SYSTEM WITH BEDDINGTON–DEANGELIS TYPE FUNCTIONAL RESPONSE

Predator–prey communities are building blocks of an ecosystem. Feeding rates reflect interference between predators in several situations, e.g. when predators form a dense colony or perform collective motion in a school, encounter prey in a region of limited size, etc. We perform spatio-temporal dynamics and pattern formation in a model aquatic system in both homogeneous and heterogeneous environments. Zooplanktons are predated by fishes and interfere with individuals of their own community. Numerical simulations are carried out to explore Turing and non-Turing spatial patterns. We also examine the effect of spatial heterogeneity on the spatio-temporal dynamics of the phytoplankton–zooplankton system. The phytoplankton specific growth rate is assumed to be a linear function of the depth of the water body. It is found that the spatio-temporal dynamics of an aquatic system is governed by three important factors: (i) intensity of interference between the zooplankton, (ii) rate of fish predation and (iii) the spatial heterogeneity. In an homogeneous environment, the temporal dynamics of prey and predator species are drastically different. While prey species density evolves chaotically, predator densities execute a regular motion irrespective of the intensity of fish predation. When the spatial heterogeneity is included, the two species oscillate in unison. It has been found that the instability observed in the model aquatic system is diffusion driven and fish predation acts as a regularizing factor. We also observed that spatial heterogeneity stabilizes the system. The idea contained in the paper provides a better understanding of the pattern formation in aquatic systems.

Propagation of Turing patterns in a plankton model

The paper is devoted to a reaction–diffusion system of equations describing phytoplankton and zooplankton distributions. Linear stability analysis of the model is carried out. Turing and Hopf stability boundaries are found. Emergence of two-dimensional spatial structures is illustrated by numerical simulations. Travelling waves between various stationary solutions are investigated. Transitions between homogeneous in space stationary solutions and Turing structures are studied.

DIFFUSIVE THREE SPECIES PLANKTON MODEL IN THE PRESENCE OF TOXIC PREY: APPLICATION TO SUNDARBAN MANGROVE WETLAND

The bloom of toxin producing phytoplankton (TPP) is an environmental issue due to its negative impact on fresh water and marine ecology. In this paper, such a phenomenon is modeled using the reaction–diffusion equations. The spatiotemporal interaction among non-toxin producing phytoplankton (NTP), TPP, and zooplankton has been considered with Holling type II and III functional responses. The stability analysis for non-spatial and spatial model system is carried out and numerical simulations are performed for a fixed set of parameter values, which is realistic to planktonic dynamics. It has been observed that on increasing the reduction rate of zooplankton, the system shows cyclic to stable behavior. The result shows that the predators which avoid to toxic prey promote the bloom. Non-Turing patchy pattern has also been observed on time evolution. In this work, we have taken the case study of Sundarban mangrove wetland which is suffering from algal bloom due to the presence of toxic Dinoflagellates and Cyanophyceae. Through the numerical simulation, it has been shown that the higher value of reduction rate of zooplankton (ξ2) is responsible for bad health of the wetland system.

Complex dynamics of diffusive predator–prey system with Beddington–DeAngelis functional response: The role of prey-taxis

An attempt has been made to understand the complex dynamics of a spatial predator–prey system with Beddington–DeAngelis type functional response in the presence of prey-taxis and subjected to homog...

A PREDATOR–PREY INTERACTION MODEL WITH SELF- AND CROSS-DIFFUSION IN AQUATIC SYSTEMS

In this paper, the complex dynamics of a spatial aquatic system in the presence of self- and cross-diffusion are investigated. Criteria for local stability, instability and global stability are obtained. The effect of critical wavelength which can drive a system to instability is investigated. We noticed that cross-diffusion coefficient can be quite significant, even for small values of off-diagonal terms in the diffusion matrix. With the help of numerical simulation, we observed the Turing patterns (spots, strips, spot-strips mixture), regular spiral patterns and irregular patchy structures. The beauty and complexity of the Turing patterns are attributed to a large variety of symmetry properties realized by different values of predators immunity, rate of fish predation and half saturation constant of predator population.

Harmful algal blooms in fresh and marine water systems: The role of toxin producing phytoplankton

In this paper, we have investigated a model with three interacting species: non-toxic phytoplankton, toxic phytoplankton and zooplankton with Holling type II and III functional responses over the space and time. The role of toxin producing phytoplankton (TPP) has been studied. We have presented the theoretical analysis of pattern formation in spatially distributed population with local diffusion. The paper highlights the heterogeneity of HABs over space and time. The choice of parameter values and the functional response is important to study the effect of TPP, also it would depend more on the nonlinearity of the system. With the help of numerical simulations, we have observed the spatial and spatiotemporal patterns for plankton system. This study demonstrates that TPP plays an important role in controlling the dynamics. We have observed that prey’s anti-predator efforts promote predator switching. It has been found that high predation of TPP helps for the coexistence of toxic, non-toxic phytoplankton and zooplankton population.

Crisis-Limited Chaotic Dynamics in an Eco-epidemiological System of the Salton Sea

In this paper, we have proposed a new eco-epidemiological model of the Salton Sea with Holling type II & IV functional responses. Numerical results are presented in the form of phase portraits, 2D scans, bifurcation analysis and basin boundary calculations. By these we have concluded that model system depicts the short-term recurrent chaos (STRC) and argued that crisis-limited chaotic dynamics can be commonly found in model eco-epidemiological systems.

Instabilities and Patterns in Zooplankton-Phytoplankton Dynamics: Effect of Spatial Heterogeneity

In this paper, we have made an attempt to understand the instabilities and complex spatiotemporal patterns induced by spatial heterogeneity in a spatial Rosenzweig - McAurthur model for phytoplankton-zooplankton-fish interaction. We have examined the effect of heterogeneity, and fish predation on the stability of the model system. Based on these conditions and by performing a series of extensive simulations, we observed the irregular patterns in the presence of spatial heterogeneity and fish predation acts as a regularizing factor. Numerical simulation results reveal that the complex temporal dynamics in natural communities may arise through the spatial dimension. Spatially induced chaos may have an important role in spatial pattern generation in heterogeneous environments.