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Highlights • A two-dimensional prey-predator model in presence of habitat complexity and prey refuge with two discrete delays has been proposed and analyzed. • The proposed model system is persistent and globally asymptotically stable if the intrinsic growth rate of prey population is greater than a threshold value, which depends upon the habitat complexity and refuge. • The negative feedback delay and gestation delay both destabilize the behavior of the system. • The numerical experiments conducted shows that the Hopf-bifurcation is supercritical, the bifurcated periodic solution is stable and its period increases. Abstract In this paper, we consider a two-dimensional prey-predator system with two delays. One delay is for negative feedback of the prey population and another is for gestation delay of the predator population. The predator is partially dependent on the prey followed by Holling type-II functional response. Due to habitat complexity and prey refuge, the Holling type-II functional response is modified in this work. We discuss the boundedness, permanence, local and global asymptotic behavior of the non-delayed and delayed models. The existence of periodic solutions via Hopf-bifurcation with respect to both the delays is established. The stability and direction of Hopf-bifurcation is also analyzed by using Normal form theory and Centre manifold theory. Lastly, numerical simulations have been carried out to confirm the analytical findings. The main objective of this work is to balance the prey-predator relationship in the presence of habitat complexity, prey refuge and delays. [ABSTRACT FROM AUTHOR]

Plankton-fish interactions are the central topic of interest related to marine ecology. Apart from the direct predation (lethal effect) of zooplankton by fish, there are some non-lethal implications in the zooplankton-fish relationship. Due to induced fear of predation, there can be a reduced reproduction rate of zooplankton species, and the effect of this non-lethal interconnection can be carried over to subsequent seasons or generations. In the current study, we tend to analyse the role of fish-induced fear in zooplankton with its carry-over effects (COEs) and a corresponding discrete delay (COE delay) in a phytoplankton-zooplankton-fish population model. We use Holling type IV and II functional responses to model the phytoplankton-zooplankton and zooplankton-fish interplay, respectively. In the well-posedness of the present biological system, firstly, we evaluate an invariant set in which the solutions of the model remain bounded. Then we prove its persistence under some ecologically well-behaved conditions. Next, we establish the conditions under which the different feasible equilibrium points exist; the existence of various interior equilibria is also set up here. To study the system's dynamical behavior, local and global stability analyses for the equilibria mentioned above are also discussed. Further, the theoretical conditions for Hopf and transcritical bifurcations in non-delayed and delayed models are determined. Impacts of non-lethal parameters, fear, and its carry-over effects, on the population densities are studied analytically and supported numerically. For intermediate values of COEs parameter, we notice that the system behaves chaotically, and decreasing (or increasing) it to low (or high) values, solution converges to interior equilibrium point through period-halving. Calculation of the largest Lyapunov exponent and drawing of Poincaré map validate the chaotic nature of the system. The chaos for medium values of COEs parameter can also be controlled by decreasing the fear parameter. Next, we numerically validate the theoretical result for transcritical bifurcation. We also note that our system shows the phenomenon of enrichment of paradox, and the attribute of multistability. In the delayed model, we observe that increasing delay can eliminate chaotic oscillations through amplitude death phenomenon. We draw various types of graphs and diagrams to assist our results. Thus we can say that the present study has various interesting characters related to non-linear models and can help biologists to study the plankton-fish models in a more detailed and pragmatic manner. [ABSTRACT FROM AUTHOR]

Oncologists and applied mathematicians are interested in understanding the dynamics of cancer-immune interactions, mainly due to the unpredictable nature of tumour cell proliferation. In this regard, mathematical modelling offers a promising approach to comprehend this potentially harmful aspect of cancer biology. This paper presents a novel dynamical model that incorporates the interactions between tumour cells, healthy tissue cells, and immune-stimulated cells when subjected to simultaneous chemotherapy and radiotherapy for treatment. We analysed the equilibria and investigated their local stability behaviour. We also study transcritical, saddle–node, and Hopf bifurcations analytically and numerically. We derive the stability and direction conditions for periodic solutions. We identify conditions that lead to chaotic dynamics and rigorously demonstrate the existence of chaos. Furthermore, we formulated an optimal control problem that describes the dynamics of tumour-immune interactions, considering treatments such as radiotherapy and chemotherapy as control parameters. Our goal is to utilize optimal control theory to reduce the cost of radiotherapy and chemotherapy, minimize the harmful effects of medications on the body, and mitigate the burden of cancer cells by maintaining a sufficient population of healthy cells. Cost-effectiveness analysis is employed to identify the most economical strategy for reducing the disease burden. Additionally, we conduct a Latin hypercube sampling-based uncertainty analysis to observe the impact of parameter uncertainties on tumour growth, followed by a sensitivity analysis. Numerical simulations are presented to elucidate how dynamic behaviour of model is influenced by changes in system parameters. The numerical results validate the analytical findings and illustrate that a multi-therapeutic treatment plan can effectively reduce tumour burden within a given time frame of therapeutic intervention. • We propose a mathematical model to assess effects of radio-chemotherapy treatments on tumour cell interaction with optimal control. • We analyse stability and bifurcations like Transcritical, Saddle node, Hopf, and GH. • Optimal control problem is formulated to minimize the tumour cells and increase the effector cells with minimum effects of drugs. • Different control techniques are compared using numerical simulation and methods like ACER and ICER to find the cost-effective strategy. • We use the LHS-PRCC method to find the global sensitivity index of the systems parameter. [ABSTRACT FROM AUTHOR]