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In the current study, we have formulated an S I epidemic model incorporating the influence of Allee effect on the dynamics of the model system in space and time. During the spread of the disease, inhibition among the susceptible and infected individuals is seen, which is measured using the Crowley–Martin type incidence function. Extensive analysis explores the system's stability and different bifurcation scenarios, including Hopf, saddle-node, transcritical, and Bogdanov–Takens. We also examine how these bifurcations react to parameter variations within the proposed model. The temporal model has further been extended to a spatial one to investigate the impact of the Allee effect on the formation of patterns for different values of the cross-diffusion coefficient. The well-posedness of the stationary solution and the global stability of the spatial system are derived. Also, the Turing instability, Hopf, and Turing bifurcation are calculated considering the transmission rate as critical parameter. In a heterogeneous environment, the spatial distribution of the susceptible population shows a complex structure with flat tabular surface and a few almost circular holes in surface plots for both strong and weak Allee effects. To further explore the role of Allee effect on pattern development for various cross-diffusion coefficient values, the temporal model has been spatially extended. Additionally, the transmission rate is taken into account while calculating the Turing instability and Hopf and Turing bifurcations. We ran numerical simulations to validate our analytical results for both spatial and nonspatial models. Our current model system can explain the recent occurrence of respiratory distress syndrome in endangered penguin species. [ABSTRACT FROM AUTHOR]

In this article, we construct an excitable memristive diffusive neuron model by considering a biophysical slow–fast bursting oscillator and study the effects of electromagnetic induction on the dynamics of the single model as well as the coupled systems. We explore various firing regimes such as tonic spiking, bursting, and mixed-mode oscillations depending on the bifurcation structure with different injected current stimuli, then perform a comparative analysis on the synchronization of the coupled oscillators by setting the model into two different network architectures. First, a diffusively coupled network is considered, and later a global network is constructed. The results suggest that the diffusively connected neurons show complete synchronization at higher couplings for bursting and tonic spiking regimes. Furthermore, we show that the extended spatial system can generate spiral-like patterns in the vicinity of a Hopf bifurcation point and observe the impact of Gaussian white noise to study its effects on pattern formation. These types of patterns are robust in the excitable model. Our results might contribute significantly to the dynamical studies of irregular neural computation. [ABSTRACT FROM AUTHOR]

In this paper, an attempt has been made to understand the role of predator's interference and additional food on the dynamics of a diffusive population model. We have studied a predator–prey interaction system with mutually interfering predator by considering additional food and Crowley–Martin functional response (CMFR) for both the reaction–diffusion model and associated spatially homogeneous system. The local stability analysis ensures that as the quantity of alternative food decreases, predator-free equilibrium stabilizes. Moreover, we have also obtained a condition providing a threshold value of additional food for the global asymptotic stability of coexisting steady state. The nonspatial model system changes stability via transcritical bifurcation and switches its stability through Hopf-bifurcation with respect to certain ranges of parameter determining the quantity of additional food. Conditions obtained for local asymptotic stability of interior equilibrium solution of temporal system determines the local asymptotic stability of associated diffusive model. The global stability of positive equilibrium solution of diffusive model system has been established by constructing a suitable Lyapunov function and using Green's first identity. Using Harnack inequality and maximum modulus principle, we have established the nonexistence of nonconstant positive equilibrium solution of the diffusive model system. A chain of patterns on increasing the strength of additional food as spots → stripes → spots has been obtained. Various kind of spatial-patterns have also been demonstrated via numerical simulations and the roles of predator interference and additional food are established. [ABSTRACT FROM AUTHOR]

In this paper, a spatial model has been designed to study a damaged diffusive eco-epidemiological system of Tilapia and Pelican populations in Salton Sea, California, USA. The nature of different equilibrium points and the existence of Hopf bifurcations are obtained. Conditions for Turing instability caused by local random movement of populations are derived. Numerically, the presence/existence of the wave of chaos phenomena is reported. Further, we show that the contact rate between susceptible and infected Tilapia population plays an important role in the distribution of the infected Tilapia population. The results suggest that the removal of infected Tilapia at regular time duration and controlling salinity will help to restore the system which provides a perspective for conservation strategy. [ABSTRACT FROM AUTHOR]

The modified three-dimensional (3D) Morris-Lecar (M-L) model is very useful to understand the spiking activities of neurons. The present article addresses the random dynamical behavior of a modified M-L model driven by a white Gaussian noise with mean zero and unit spectral density. The applied stimulus can be expressed as a random term. Such random perturbations are represented by a white Gaussian noise current added through the electrical potential of membrane of the excitatory principal cells. The properties of the stochastic system (perturbed one) and noise induced mixed mode oscillation are analyzed. The Lyapunov spectrum is computed to present the nature of the system dynamics. The noise intensity is varied while keeping fixed the predominant parameters of the model in their ranges and also observed the changes in the dynamical behavior of the system. The dynamical synchronization is studied in the coupled M-L systems interconnected by excitatory and inhibitory neurons with noisy electrical coupling and verified with similarity functions. This result suggests the potential benefits of noise and noise induced oscillations which have been observed in real neurons and how that affects the dynamics of the neural model as well as the coupled systems. The analysis reports that the modified M-L system which has the limit cycle behavior can show a type of phase locking behavior which follows either period adding (i.e. 1:1, 2:1, 3:1, 4:1) sequences or Farey sequences. For the coupled neural systems, complete synchronization is shown for sufficient noisy coupling strength. [ABSTRACT FROM AUTHOR]

Recently, the 2014 Ebola virus (EBOV) outbreak in West Africa was the largest outbreak to date. In this paper, an attempt has been made for modeling the virus dynamics using an SEIR model to better understand and characterize the transmission trajectories of the Ebola outbreak. We compare the simulated results with the most recent reported data of Ebola infected cases in the three most affected countries Guinea, Liberia and Sierra Leone. The epidemic model exhibits two equilibria, namely, the disease-free and unique endemic equilibria. Existence and local stability of these equilibria are explored. Using central manifold theory, it is established that the transcritical bifurcation occurs when basic reproduction number passes through unity. The proposed Ebola epidemic model provides an estimate to the potential number of future cases. The model indicates that the disease will decline after peaking if multisectorial and multinational efforts to control the spread of infection are maintained. Possible implication of the results for disease eradication and its control are discussed which suggests that proper control strategies like: (i) transmission precautions, (ii) isolation and care of infectious Ebola patients, (iii) safe burial, (iv) contact tracing with follow-up and quarantine, and (v) early diagnosis are needed to stop the recurrent outbreak. [ABSTRACT FROM AUTHOR]

In this paper, an eco-epidemiological model in which both species diffuse along a spatial gradient has been shown to exhibit temporal chaos at a fixed point in space. The proposed model is a modification of the model recently presented by Upadhyay and Roy [2014]. The spatial interactions among the species have been represented in the form of reaction-diffusion equations. The model incorporates the intrinsic growth rate of fish population which varies linearly with the depth of water. Numerical results show that diffusion can drive otherwise stable system into aperiodic behavior with sensitivity to initial conditions. We show that spatially induced chaos plays an important role in spatial pattern formation in heterogeneous environment. Spatiotemporal distributions of species have been simulated using the diffusivity assumptions realistic for natural eco-epidemic systems. We found that in heterogeneous environment, the temporal dynamics of both the species are drastically different and show chaotic behavior. It was also found that the instability observed in the model is due to spatial heterogeneity and diffusion-driven. Cumulative death rate of predator has an appreciable effect on model dynamics as the spatial distribution of all constituent populations exhibit significant changes when this model parameter is changed and it acts as a regularizing factor. [ABSTRACT FROM AUTHOR]

In this paper, an attempt has been made to study the spatial and temporal dynamical interactions among the species of wetland ecosystem through a mathematical model. The model represents the population dynamics of phytoplankton, zooplankton and fish species found in Chilika lake, Odisha, India. Nonlinear stability analysis of both the temporal and spatial models has been carried out. Maximum sustainable yield and optimal harvesting policy have been studied for a nonspatial model system. Numerical simulation has been performed to figure out the parameters responsible for the complex dynamics of the wetland system. Significant outcomes of our numerical findings and their interpretations from an ecological point of view are provided in this paper. Numerical simulation of spatial model exhibits some interesting and beautiful patterns. We have also pointed out the parameters that are responsible for the good health of wetland ecosystem. [ABSTRACT FROM AUTHOR]

In this paper, we have proposed and analyzed a simple model of Influenza spread with an asymptotic transmission rate. Existence and uniqueness of solutions are established and shown to be uniformly bounded for all non-negative initial values. We have also found a sufficient condition which ensures the persistence of the model system. This implies that both susceptible and infected will always coexist at any location of the inhabited domain. This coexistence is independent of values of the diffusivity constants for two subpopulations. The global stability of the endemic equilibrium is established by constructing a Lyapunov function. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation, conditions for Hopf and Turing bifurcations are obtained. We have also studied the criteria for diffusion-driven instability caused by local random movements of both susceptible and infective subpopulations. Turing patterns selected by the reaction-diffusion system under zero flux boundary conditions have been explored. Numerical simulations show that contact rate, β which is related to the reproduction number , plays an important role in spatial pattern formation. It was found that diffusion has appreciable influence on spatial spread of epidemics. The wave of chaos appears to be a dominant mode of disease dispersal. This suggests a bidirectional spread for influenza epidemics. The epidemic propagates in the form of nonchaotic and chaotic waves as observed in H1N1 incidence data of positive tests in 2009 in the United States. We have conducted numerical simulations to confirm the analytic work and observed interesting behaviors. This suggests that influenza has a complex dynamics of spatial spread which evolves with time. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We examine and assess deterministic chaos as an observable. First, we present the development of model ecological systems. We illustrate how to apply the Kolmogorov theorem to obtain limits on the parameters in the system, which assure the existence of either stable equilibrium point or stable limit cycle behavior in the phase space of two-dimensional (2D) dynamical systems. We also illustrate the method of deriving conditions using the linear stability analysis. We apply these procedures on some basic existing model ecological systems. Then, we propose four model ecological systems to study the dynamical chaos (chaos and intermittent chaos) and cycles. Dynamics of two predation and two competition models have been explored. The predation models have been designed by linking two predator–prey communities, which differ from one another in one essential way: the predator in the first is specialist and that in the second is generalist. The two competition models pertain to two distinct competition processes: interference and exploitative competition. The first competition model was designed by linking two predator–prey communities through inter-specific competition. The other competition model assumes that a cycling predator–prey community is successfully invaded by a predator with linear functional response and coexists with the community as a result of differences in the functional responses of the two predators. The main criterion behind the selection of these two model systems for the present study was that they represent diversity of ecological interactions in the real world in a manner which preserves mathematical tractability. For investigating the dynamic behavior of the model systems, the following tools are used: (i) calculation of the basin boundary structures, (ii) performing two-dimensional parameter scans using two of the parameters in the system as base variables, (iii) drawing the bifurcation diagrams, and (iv) performing time series analysis and drawing the phase space diagrams. The results of numerical simulation are used to distinguish between chaotic and cyclic behaviors of the systems. The conclusion that we obtain from the first two model systems (predation models) is that it would be difficult to capture chaos in the wild because ecological systems appear to change their attractors in response to changes in the system parameters quite frequently. The detection of chaos in the real data does not seem to be a possibility as what is present in ecological systems is not robust chaos but short-term recurrent chaos. The first competition model (interference competition) shares this conclusion with those of predation ones. The model with exploitative competition suggests that deterministic chaos may be robust in certain systems, but it would not be observed as the constituent populations frequently execute excursions to extinction-sized densities. Thus, no matter how good the data characteristics and analysis techniques are, dynamical chaos may continue to elude ecologists. On the other hand, the models suggest that the observation of cyclical dynamics in nature is the most likely outcome. [ABSTRACT FROM AUTHOR]