The declaration of a nationwide lockdown in India led to millions of migrant workers, particularly from Uttar Pradesh (UP) and Bihar, returning to their home states without proper transportation and social distancing from cities such as Delhi, Mumbai, and Hyderabad. This unforeseen migration and social mixing accelerated the transmission of diseases across the country. To analyze the impact of reverse migration on disease progression, we have developed a disease transmission model for the neighboring Indian states of Delhi and UP. The model’s essential mathematical properties, including positivity, boundedness, equilibrium points (EPs), and their linear stability, as well as computation of the basic reproduction number ( R 0 ) \left({R}_{0}) , are studied. The mathematical analysis reveals that the model with active reverse migration cannot reach a disease-free equilibrium, indicating that the failure of restrictive mobility intervention caused by reverse migration kept the disease propagation alive. Further, PRCC analysis highlights the need for effective home isolation, better disease detection techniques, and medical interventions to curb the spread. The study estimates a significantly shorter doubling time for exponential growth of the disease in both regions. In addition, the occurrence of synchronous patterns between epidemic trajectories of the Delhi and UP regions accentuates the severe implications of migrant plight on UP’s already fragile rural health infrastructure. By using COVID-19 incidence data, we quantify key epidemiological parameters, and our scenario analyses demonstrate how different lockdown plans might have impacted disease prevalence. Based on our observations, the transmission rate has the most significant impact on COVID-19 cases. This case study exemplifies the importance of carefully considering these issues before implementing lockdowns and social isolation throughout the country to combat future outbreaks.

This work investigates synchronization within a system of two coupled food chains. We assess the scenario if and how the chaotic dynamics of a food chain can be suppressed in the presence of other food chains and order can be restored within the system of coupled food chains. For this, we consider the extended version of Hasting Powell model (eHPM), in which two bio-controlling parameters, refugia, and Allee, are already present to suppress chaos. Through imperative mathematical properties and numerical simulations of the system, we first understand the impact of the interplay of refugia and Allee effects in defining the qualitative behaviour of the system. After analyzing the system's sensitive dependence on the initial condition, we use the master-slave scheme to attain synchronization within the system of two coupled food chains defined for three different frameworks. Following the active control method, Lyapunov stability theory is used to deduce the analytical expression for nonlinear active controllers for the synchronization of three different frameworks namely - (a) two identical eHPMs but with different qualitative behaviour (master eHPM possesses regular whereas slave eHPM possesses chaotic dynamics), (b) non-identical eHPM (master) and HP model (slave)~( eHPM possesses regular whereas HP model possesses chaotic dynamics), and (c) non-identical eHPM (master) and HP model (slave) with different initial conditions (both eHPM and HP model posses chaotic dynamics). Using techniques from active control theory, it is demonstrated that all three proposed systems can be phase and chaos synchronized. The time series plots of the error dynamics are used to validate that the slave system from all three frameworks adheres to the same control functions effectively for the synchronization process. Numerical simulation from the study can provide basis for assessing the potential significance of synchronization in a network of coupled food chains where Refugia and Allee effects are present.

Seasonal effects powerfully shape the population dynamics with periodic climate changes because species naturally adjust their dynamics with seasonal variations. In response to these effects, sometimes population dynamics exhibit synchrony or generate chaos. However, synchronized dynamics enhance species’ persistence in naturally unstable environments; thus, it is imperative to identify parameters that alter the dynamics of an ecosystem and bring it into synchrony. This study examines how ecological parameters enable species to adapt their dynamics to seasonal changes and achieve phase synchrony within ecosystems. For this, we incorporate seasonal effects as a periodic sinusoidal function into a tri-trophic food chain system where two crucial bio-controlling parameters, Allee and refugia effects, are already present. First, it is shown that the seasonal effects disrupt the limit cycle and bring chaos to the system. Further, we perform rigorous mathematical analysis to perform the dynamical and analytical properties of the nonautonomous version of the system. These properties include sensitive dependence on initial condition (SDIC), sensitivity analysis, bifurcation results, the positivity and boundedness of the solution, permanence, ultimate boundedness, and extinction scenarios of species. The SDIC characterizes the presence of chaotic oscillations in the system. Sensitivity analysis determines the parameters that significantly affect the outcome of numerical simulations. The bifurcation study concerning seasonal parameters shows a higher dependency of species on the frequency of seasonal changes than the severity of the season. The bifurcation study also examines the bio-controlling parameters and reveals various dynamic states within the system, such as fold, transcritical branch points, and Hopf points. Moreover, the mathematical analysis of our seasonally perturbed system reveals the periodic coexistence of all species and a globally attractive solution under certain parametric constraints. Finally, we examine the role of essential parameters that contribute to phase synchrony. For this, we numerically investigate the defining role of the coupling dimension coefficient, bio-controlling parameters, and other parameters associated with seasonality. This study infers that species can tune their dynamics to seasonal effects with low seasonal frequency, whereas the species’ tolerance for the severity of seasonal effects is relatively high. The research also sheds light on the correlation between the degree of phase synchrony, prey biomass levels, and the severity of seasonal forcing. This study offers valuable insights into the dynamics of ecosystems affected by seasonal perturbations, with implications for conservation and management strategies.

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