Mathematical modeling of cascading migration in a tri-trophic food-chain system

Daylight driven vertical migration of mesozooplankton in the Arabian sea and the Bay of Bengal

Diurnal vertical migration (DVM) of mesozooplankton in the Deep Scattering Layer (DSL) of the Indian seas is poorly studied. This cyclical vertical migration substantially controls the carbon sequestration in the ocean. The present research is a comprehensive examination to analyse the factors affecting the DVM pattern of the zooplankton community in the Arabian Sea (AS) and the Bay of Bengal (BoB). Echo sounder profiling was conducted at shallow depths (∼10-400m) of the AS (January 2023) and BoB (March 2023) with a period of 24 h to monitor the DVM pattern of the DSL. Vertical migration in both basins showcased the notable influence of the spatio-temporal contrast in the occurrence of daybreak, with the day (descend) and night (ascend) cycle of the DSL. Delayed descent was observed in the AS contrary to BoB, owing to the delayed day break in the AS relative to BoB. Intensity and temporal pattern of the incoming solar radiation were correlated with the DVM whereas diurnal variation of sea surface temperature was observed to be contrasting. The preliminary analysis is indicative of the diversified community structure of the zooplankton community in these basins resulting from the vertical migration. Furthermore, it is conclusive that the surface residence time of the zooplankton is distinct and is affirmed based on daybreak and light intensity particular for each basin. Since daybreak vary with the geolocation, sole dependence on a particular time for migration study can be erroneous, which is highlighted in the present study.

Chaos in a nonautonomous eco-epidemiological model with delay

In this paper, we propose and analyze a nonautonomous predator-prey model with disease in prey, and a discrete time delay for the incubation period in disease transmission. Employing the theory of differential inequalities, we find sufficient conditions for the permanence of the system. Further, we use Lyapunov's functional method to obtain sufficient conditions for global asymptotic stability of the system. We observe that the permanence of the system is unaffected due to presence of incubation delay. However, incubation delay affects the global stability of the positive periodic solution of the system. To reinforce the analytical results and to get more insight into the system's behavior, we perform some numerical simulations of the autonomous and nonautonomous systems with and without time delay. We observe that for the gradual increase in the magnitude of incubation delay, the autonomous system develops limit cycle oscillation through a Hopf-bifurcation while the corresponding nonautonomous system shows chaotic dynamics through quasi-periodic oscillations. We apply basic tools of non-linear dynamics such as Poincaré section and maximum Lyapunov exponent to confirm the chaotic behavior of the system.

THE ROLE OF ADDITIONAL FOOD IN A PREDATOR–PREY MODEL WITH A PREY REFUGE

In this paper, we propose and analyze a predator–prey model with a prey refuge and additional food for predators. We study the impact of a prey refuge on the stability dynamics, when a constant proportion or a constant number of prey moves to the refuge area. The system dynamics are studied using both analytical and numerical techniques. We observe that the prey refuge can replace the predator–prey oscillations by a stable equilibrium if the refuge size crosses a threshold value. It is also observed that, if the refuge size is very high, then the extinction of the predator population is certain. Further, we observe that enhancement of additional food for predators prevents the extinction of the predator and also replaces the stable limit cycle with a stable equilibrium. Our results suggest that additional food for the predators enhances the stability and persistence of the system. Extensive numerical experiments are performed to illustrate our analytical findings.

Chaos to order — Effect of random predation in a Holling type IV tri-trophic food chain system with closure terms

Complex dynamics of modified Hastings–Powell (HP) model (phytoplankton-zoo-plankton-fish) with Holling type IV functional response and density-dependent mortality (closure terms) for top predator species is investigated in this paper. Closure terms describe the mortality of top predator in plankton food chain models. Modified HP model with Holling type IV functional response gives rise to similar type of chaotic dynamics (inverted “teacup attractor”) as observed in original HP model with Holling type II functional response. It is observed that introduction of nonlinear closure terms eliminate chaos and system dynamics becomes stable. Observation of this paper support the “Steele–Henderson conjecture” that, nonlinear closure terms eliminate or reduces limit cycles and chaos in plankton food chain models. Chaotic or stable dynamics are numerically verified by Lyapunov exponents (LE) method and Sil’nikov eigenvalue analysis and also illustrated graphically by plotting bifurcation diagrams. It is assumed that mortality of fish population, caused by higher-order predators (which are not explicitly included in the model) is not constant, rather it exhibits random variation throughout the year. To incorporate the effect of random mortality of fish population, white noise term is introduced into the original deterministic model. It is observed that the corresponding stochastic model is stable in mean square when the intensity of noise is small.

Chaos control of Hastings-Powell model by combining chaotic motions

In this paper, we propose a Parameter Switching (PS) algorithm as a new chaos control method for the Hastings-Powell (HP) system. The PS algorithm is a convergent scheme that switches the control parameter within a set of values while the controlled system is numerically integrated. The attractor obtained with the PS algorithm matches the attractor obtained by integrating the system with the parameter replaced by the averaged value of the switched parameter values. The switching rule can be applied periodically or randomly over a set of given values. In this way, every stable cycle of the HP system can be approximated if its underlying parameter value equalizes the average value of the switching values. Moreover, the PS algorithm can be viewed as a generalization of Parrondo's game, which is applied for the first time to the HP system, by showing that losing strategy can win: "losing + losing = winning." If "loosing" is replaced with "chaos" and, "winning" with "order" (as the opposite to "chaos"), then by switching the parameter value in the HP system within two values, which generate chaotic motions, the PS algorithm can approximate a stable cycle so that symbolically one can write "chaos + chaos = regular." Also, by considering a different parameter control, new complex dynamics of the HP model are revealed.