Nonautonomous Model Research Articles

Dynamics of autonomous and nonautonomous Leslie–Gower model for the impacts of fear and its carry‐over effects including predator harvesting

Recently, certain biological field experiments and research findings have revealed that predators not only reduce prey populations through direct predation (lethal effect) but also indirectly affect the growth rate of prey by inducing fear; these nonlethal effects can be carried over seasons or generations. In this paper, we proposed an autonomous and a nonautonomous Leslie–Gower model incorporating some biological factors such as fear and its carry‐over effects, prey refuge, and nonlinear predator harvesting. In the autonomous model, first, we examined the well‐posedness, positivity solutions, and their boundedness. Also, it is shown that new equilibrium points emerge and disappear with the change in intrinsic growth rate of predator. Further, we compute and analyzed the local and global stability at the interior equilibrium points. Furthermore, Hopf bifurcation and direction and stability of limit cycle at interior equilibrium points are established. Moreover, the sensitivity analysis of the biological parameters is carried out numerically with two statistical methods via Latin hypercube sampling and partial rank correlation coefficients. It was found that at the small values of fear factor, carry‐over effect, and refuge behavior of prey parameters, the autonomous system exhibits oscillatory behavior, whereas for small values of prey birth rate and harvesting effort, the system remains stable. However, when intrinsic growth rate of predator increases from a low value to a higher value, the system shows transition from stability to instability and back to stability. We also explore the effects of seasonal variations in biological parameters in the nonautonomous model. Additionally, we investigate the theoretical existence of positive periodic solutions using the continuation theorem. The influences of seasonal variations in some parameters such as prey's birth rate, fear, and its carry‐over effects; refuge behavior of prey; intrinsic growth rate of predator; and harvesting effort are then numerically investigated. The results reveal the emergence of positive periodic solutions and bursting patterns in the seasonally forced model.

Chaos and extinction risks of sexually reproductive generalist top predator in a seasonally forced food chain system with Allee effect.

This paper investigates the dynamics of a tritrophic food chain model incorporating an Allee effect, sexually reproductive generalist top predators, and Holling type IV and Beddington-DeAngelis functional responses for interactions across different trophic levels. Analytically, we explore the feasible equilibria, their local stability, and various bifurcations, including Hopf, saddle-node, transcritical, and Bogdanov-Takens bifurcations. Numerical findings suggest that higher Allee intensity in prey growth leads to the inability of species coexistence, resulting in a decline in species density. Likewise, a lower reproduction rate and a higher strength of intraspecific competition among top predators also prevent the coexistence of species. Conversely, a rapid increase in the reproduction rate and a decrease in the strength of intraspecific competition among top predators enhance the densities of prey and top predators while decreasing intermediate predator density. We also reveal the presence of bistability and tristability phenomena within the system. Furthermore, we extend our autonomous model to its nonautonomous counterpart by introducing seasonally perturbed parameters. Numerical analysis of the nonautonomous model reveals that higher seasonal strength in the reproduction rate and intraspecific competition of top predators induce chaotic behavior, which is also confirmed by the maximum Lyapunov exponent. Additionally, we observe that seasonality may lead to the extinction of species from the ecosystem. Factors such as the Allee effect and growth rate of prey can cause periodicity in population densities. Understanding these trends is critical for controlling changes in population density within the ecosystem. Ecologists, environmentalists, and policymakers stand to benefit significantly from the invaluable insights garnered from this study. Specifically, our findings offer pivotal guidance for shaping future strategies aimed at safeguarding biodiversity and maintaining ecological stability amidst changing environmental conditions. By contributing to the existing body of knowledge, our study advances the field of ecological science, enhancing the comprehension of predator-prey dynamics across diverse ecological conditions.

Diffusion and distributed delay effects in a predator–prey system: A mathematical analysis

This study explores a nonautonomous model of prey-predator dynamics, featuring continuous distributed delay and diffusion, with time-dependent parameters. Our model depicts two ecological patches hosting two species: predators remain localized in one patch, while prey can disperse between both. Our analysis concentrates on ensuring the model’s consistent presence and achieving global stability. Additionally, we derive necessary conditions for periodic solutions by developing a pertinent Lyapunov function.

MATHEMATICAL MODEL OF MEASLES IN TURKEY

In this paper, we use a previously developed measles model to forecast measles in Turkey for the period 1970–2021. We study the structural identifiability of the model both by hand and using software. By hand, we assume the prevalence and the total population size are given. Using software, we assume the incidence and the total population size are given. The model is structurally identifiable if one of the three parameters is fixed. We notice that Turkey has a significant change in time of the immigration rate and vaccination proportions, so we assume these two quantities are time-dependent. We fit the nonautonomous model to the measles incidences in Turkey for 1970–2021. We perform practical identifiability of the fitted model, and find that all parameters but one are practically identifiable. When fixing the unidentifiable parameter to a value derived from additional data, we obtain that all parameters are practically identifiable.

Dynamical behaviors of autonomous and nonautonomous models of generalist predator–prey system with fear, mutual interference and nonlinear harvesting

In this study, we explore a predator–prey model to evaluate the impacts of fear, mutual interference, and harvesting, as these factors play crucial roles in modifying population dynamics, trophic cascades, and species interactions. Additionally, we enhance the model to incorporate the seasonality of specific ecological factors across time. The autonomous and associated nonautonomous models have undergone thorough mathematical analysis and extensive numerical simulations in the course of the study. We conduct a fundamental differential sensitivity analysis to discern the key parameters influencing both prey and predator populations within the ecosystem. Through numerical investigations of the autonomous model, we notice that in the presence (absence) of harvesting, the mutual interference among predators destabilizes (stabilizes) the coexisting equilibrium. The fear effect significantly reduces the densities of prey and predator populations, and destabilizes the coexisting equilibrium by producing multiple stability switches. The survival of predators substantially depends on the focal prey if their growth from other food sources is relatively low. Notably, the harvesting of predators is beneficial and promotes the coexistence of species only when their growth from other food sources is too much. Moreover, the system shows bistability, tristability, and multiple limit cycles under identical ecological conditions. We also show various phenomena that manifest within the seasonally forced model, including the emergence of periodic solutions, higher periodic solutions, bursting patterns, and chaotic dynamics. Remarkably, the influence of seasonal forcing possesses the capability to govern the chaotic dynamics by giving rise to an exceedingly quasi-periodic arrangement in the prey and predator densities.

Multi-dimensional chaos initiated by short pulses in non-autonomous radio-physical generator

A non-autonomous model of the Anishchenko–Astakhov generator in the regime of periodic and chaotic self-oscillations is considered. A periodic sequence of short pulses is considered as an external force. It is shown that the synchronization picture is close in structure to the classical synchronization picture observed in a two-dimensional system, but the pulse action leads to the excitation of chaotic oscillations, including those characterized by a different spectrum of Lyapunov exponents. In particular, it is shown appearance of hyperchaos and chaos with additional close to zero Lyapunov exponent. Phenomenological scenarios for the development of multi-dimensional chaos related to destruction of two-frequency tori are described. Hyperchaos is formed via hierarchy of discrete Shilnikov attractors arise as a result of sequence of Neimark-Sacker bifurcations. Chaos with additional close to zero Lyapunov exponent occurs as impact of saddle tori appeared via sequence of torus-doubling bifurcations.

A nonautonomous model for the interaction between a size-structured consumer and an unstructured resource.

In this paper, we propose and analyze a nonautonomous model that describes the dynamics of a size-structured consumer interacting with an unstructured resource. We prove the existence and uniqueness of the solution of the model using the monotone method based on a comparison principle. We derive conditions on the model parameters that result in persistence and extinction of the population via the upper-lower solution technique. We verify and complement the theoretical results through numerical simulations.

A nonautonomous model for the effect of rarity value on the dynamics of a predator–prey system with variable harvesting

The human inclination to value rarity contributes to the exploitation of many wildlife species by driving up their market value when their numbers are scarce. This research investigates a model to explore how the value attached to rarity impacts the harvesting of predator species and subsequently influences the dynamics of predator–prey interactions in an ecological system. To broaden the model's scope, it's extended to its nonautonomous form by allowing the harvesting rate of predator species and the degree of neutrality in consumers to respond for the rarity of predator species to change over time. Both the autonomous and nonautonomous systems are explored through analytical and numerical investigation. Numerical findings reveal that as the prey population's growth rate gradually increases, the unforced system shifts from stable boundary equilibrium (harvesting absent) to stable interior equilibrium, then to oscillatory interior equilibrium, followed by period doubling oscillations leading to chaotic behavior. The autonomous system exhibits limit cycle oscillations around the boundary equilibrium when predator species are minimally harvested. Moderate harvesting stabilizes the system, while excessive harvesting induces instability. However, the chaotic oscillations can be eliminated through period‐halving route upon decreasing the consumers' response to predator rarity. Furthermore, the seasonally forced model demonstrates periodic solutions, higher order periodic solutions, and chaotic dynamics contingent upon different settings of prey growth rate, the strength of seasonality in predator harvesting, and the consumers' response to predator rarity.

Dynamics of a stoichiometric phytoplankton-zooplankton model with season-driven light intensity.

Chemical heterogeneity significantly influences the dynamics of phytoplankton and zooplankton interactions through its effects on phytoplankton carrying capacity and zooplankton ingestion rates. Our central objective of this study was to develop and examine a nonautonomous model of phytoplankton-zooplankton growth, which incorporates season-driven variations in light intensity and chemical heterogeneity. The dynamics of the system is characterized by positive invariance, dissipativity, boundary dynamics, and internal dynamics. Subsequently, numerical simulations were conducted to validate the theoretical findings and to elucidate the effects of seasonal light intensity, nutrient availability, and zooplankton loss rates on phytoplankton dynamics. The outcomes of our model and analysis offer a potential explanation for seasonal phytoplankton blooms.

Mathematical modeling of the infectious spread and outbreak dynamics of avian influenza with seasonality transmission for chicken farms

A compartmental model with a time-varying contact rate, the seasonality effect, and its corresponding nonautonomous model are investigated. The model is developed based on the six compartments: susceptible, latent, infected, asymptomatic, treated, and recovered individuals. We determine the effective reproduction number for this nonautonomous system, and analytic discussion shows that at least one positive periodic solution exists for R0>1. The model is simulated using the RK-45 numerical method, and the parameter values for the model are taken from the available literature. From the numerical results, we observe that the degree of seasonality and vaccine efficacy significantly impact the amplitude of the epidemic curve. The latent-infected phase plane shows that periodic solutions exhibit a period-doubling bifurcation as the amplitude of seasonality increases. Finally, the model outcome was compared with the actual field data and found to be consistent.

Dynamics of a modified Leslie–Gower Holling-type II eco-epidemiological model on time scales

Migratory birds play an important role in the transmission of illnesses like West Nile Virus and Salmonella, etc. The long-time dynamic behaviors of a novel class of nonautonomous model for migratory birds with Leslie–Gower Holling-type II functional response and saturation incidence rate on time scales are investigated in this study. Within this scope, certain necessary conditions are found for the persistence and extinction of the illness under very weak assumptions. Furthermore, the model's global attractiveness is examined by creating a Lyapunov function. Additionally, included are numerical simulations ( T = R , Z ) that back up our theoretical research.

Modeling Typhoid Fever Dynamics: Stability Analysis and Periodic Solutions in Epidemic Model with Partial Susceptibility

Mathematical models play a crucial role in predicting disease dynamics and estimating key quantities. Non-autonomous models offer the advantage of capturing temporal variations and changes in the system. In this study, we analyzed the transmission of typhoid fever in a population using a compartmental model that accounted for dynamic changes occurring periodically in the environment. First, we determined the basic reproduction number, R0, for the periodic model and derived the time-average reproduction rate, [R0], for the non-autonomous model as well as the basic reproduction number, R0A, for the autonomous model. We conducted an analysis to examine the global stability of the disease-free solution and endemic periodic solutions. Our findings demonstrated that when R0<1, the disease-free solution was globally asymptotically stable, indicating the extinction of typhoid fever. Conversely, when R0>1, the disease became endemic in the population, confirming the existence of positive periodic solutions. We also presented numerical simulations that supported these theoretical results. Furthermore, we conducted a sensitivity analysis of R0A, [R0] and the infected compartments, aiming to assess the impact of model parameters on these quantities. Our results showed that the human-to-human infection rate has a significant impact on the reproduction number, while the environment-to-human infection rate and the bacteria excretion rate affect long-cycle infections. Moreover, we examined the effects of parameter modifications and how they impact the implementing of efficient control strategies to combat TyF. Although our model is limited by the lack of precise parameter values, the qualitative results remain consistent even with alternative parameter settings.

A non-autonomous approach to study the impact of environmental toxins on nutrient-plankton system

The interaction between phytoplankton and zooplankton has a significant impact on the marine ecology. The interplay between these two species is the building blocks for most of the food webs operating in an aquatic ecosphere. The environmental toxins released by different external sources also affect the phytoplankton-zooplankton dynamics. In the present study, we propose a model to explore the kinetics of a nutrient-phytoplankton-zooplankton-environmental toxins (NPZT) system. The defence mechanism of phytoplankton against zooplankton is reflected through modified Holling type IV response, whereas the consumption of nutrients by phytoplankton is outlined by Holling type II response. The external toxins are assumed to have the capability of reducing the birth rate of phytoplankton species after coming into contact with their cells. To make our model more pragmatic, seasonal variation in the parameters is also taken into account. Firstly, we do the analysis related to the autonomous model (non-seasonal) like; its boundedness, existence of equilibrium points, their stability analysis, and occurrence of Hopf-bifurcation. Further, for the non-autonomous model (seasonal), we analyze the existence of positive periodic solution and its global stability. Through numerical simulations, we observe that for the non-seasonal model, increasing the rate of suppressing phytoplankton’s growth by environmental toxin, and rate at which environmental toxin is added to system make it unstable through Hopf-bifurcation. These oscillations can be removed by raising phytoplankton’s inhibitory effect against zooplankton, and this increment also leads to the extinction of the zooplankton population, making zooplankton free equilibrium a stable one. Both models, non-seasonal as well as seasonal manifest different types of multistability, and this is an exciting character associated with non-linear models. We also note that the inclusion of seasonality in our system promotes the coexistence of all populations. Further, through numerical simulations, we show that making some of the parameters seasonal can cause the emergence of chaos in the system. To verify chaos, we sketch the Poincaré map and evaluate the maximum Lyapunov exponent. The seasonal model also shows the switching of stability through different periodic and chaotic windows on varying the maximum intrinsic growth rate for phytoplankton, and contact rate between environmental toxin and phytoplankton. To substantiate our results, we picture several time-series graphs, basins of attraction, one and two-parametric bifurcation diagrams. Thus we expect that the present work can assist biologists and mathematicians in studying nutrient-plankton systems in a more detailed and realistic manner. This study can also help researchers in the estimation of non-seasonal as well as seasonal parameters while studying these types of complex non-linear models. Therefore, the present work seems to be enriched from a mathematical and biological point of view.

Persistence criteria for a chemostat with variable nutrient input and variable washout with delayed response in growth

We study a general single-species chemostat model with non-autonomous input and washout, and a discrete-time delay between consumption of the nutrient and growth. The goal of this article is to provide sufficient and necessary conditions for persistence. The persistence criteria obtained here extend previous works done for the autonomous model with delay and for the non-autonomous model without delay. Furthermore, in the particular case of periodic nutrient input and the same periodic washout with delayed response in growth, the persistence criteria can be expressed in terms of the average of the involved functions. We remark in no case we need to impose any restrictions on the size of the delay.

Surface Temperature Influences the Population of Limnothrissa miodon in Lake Kariba

Global warming is a serious world problem where earth’s temperature has been reported to increase over the years; the aquatic ecosystems are also not the exceptions. But, the effects of this phenomenon on the aquatic ecosystems are not well understood. This study aims to understand the influence of surface temperature on the population density of Limnothrissa miodon in Lake Kariba. We constructed a mathematical model on the population dynamics of Limnothrissa miodon with nutrients, phytoplankton, zooplankton, and Hydrocynus vittatus. Lake surface water temperature was modelled by a cosine function, and the parameters were estimated from data fitting. Numerical simulations were used to determine the stability of the nonautonomous model. Numerical simulation results of the nonautonomous model showed a stable periodic orbit for varying initial conditions, and therefore, instability. Numerical techniques were used to investigate the influence of surface water temperature on Limnothrissa miodon. Results from the model with fitted lake surface water temperature data showed that a shift in the optimal temperature for phytoplankton growth from 25 ° C to 34 ° C, corresponding to dominance of Cyanophyceae over Chlorophyceae, resulted in a decline in the population density of Limnothrissa miodon. Numerical results showed that the population density of Limnothrissa miodon declines after an optimum temperature of 30 ° C for phytoplankton growth. Numerical simulation results suggested that warming of the lake may lead to a decline in Limnothrissa miodon population density in Lake Kariba.

Dynamics of Predator-prey Model under Fluctuation Rescue Effect

This paper presents a novel idea as it investigates the rescue effect of the prey with fluctuation effect for the first time to propose a modified predator-prey model that forms a non-autonomous model. However, the approximation method is utilized to convert the non-autonomous model to an autonomous one by simplifying the mathematical analysis and following the dynamical behaviors. Some theoretical properties of the proposed autonomous model like the boundedness, stability, and Kolmogorov conditions are studied. This paper's analytical results demonstrate that the dynamic behaviors are globally stable and that the rescue effect improves the likelihood of coexistence compared to when there is no rescue impact. Furthermore, numerical simulations are carried out to demonstrate the impact of the fluctuation rescue effect on the dynamics of the non-autonomous model. The analytical and numerical results show a more coexisted model between prey and predator, which can help any extinction-threatened ecosystem.

Pullback attractor for a class of non-autonomous lattice differential equations with delays

We investigate the dynamics of a class of lattice differential equations with delays in arbitrary spatial dimensions. We prove the existence of a unique solution for the non-autonomous model and the existence of a pullback attractor for the associated process. The class we study includes examples of discrete models with delays not considered before.

Large time behavior of nonautonomous differential systems modeling antibiotic‐resistant bacteria in rivers

In this paper, we study a general nonautonomous model for bacterial dynamics in rivers. The mathematical model is represented by a nonautonomous system of nonlinear ordinary differential equations. We show the existence of a bounded positive invariant and attracting set. By using the Lyapunov function method, we establish global stability of steady‐state solutions of the associated autonomous system. Second, the existence of positive periodic solutions of the nonautonomous system is proven using a continuation theorem based on coincidence degree theory.

A periodic trachoma epidemic model on heterogeneous networks

In this paper, considering the impact of environmental change on trachoma transmission, we establish a periodic trachoma epidemic model on heterogeneous networks, which is a non-autonomous model. We prove theoretically the existence of basic reproduction number shown by R0. And then, we demonstrate that the disease-free periodic solution is globally asymptotically stable if R0<1, and there exists a unique positive periodic solution which is also is globally asymptotically stable if R0>1. The method obtained in this paper can be seen as an extension for proving the global stability of the high-dimensional periodic epidemic model on heterogeneous networks. And our results provide theoretical basis for decision-makers to make control strategies.

Chaos in a nonautonomous model for the impact of media on disease outbreak

Social media initiatives educate the public about the tools and strategies necessary to defend themselves from emerging diseases. The effect of media advertisements in combating an infectious disease is explored in the present investigation. The numerical results show that the system destabilizes when the growth rate of media advertisements goes above a certain value. For lower ranges of the rate of distribution of awareness through social media platforms, our system showcases stable endemic state. But as the rate of disseminating awareness surpasses a critical value, the system undergoes a Hopf bifurcation and limit cycle oscillations appear. However, the persistent oscillations are killed out and the system returns to a stable endemic state as the rate of distribution of awareness exceeds another threshold value. Extremely large values of the awareness distribution through media advertisements eradicate the disease from the entire community. Model parameters with significant impacts on the disease prevalence and eradication are identified through sensitivity analysis. We extend our model to encapsulate the seasonal variation in the rate of media advertisements. Sufficient conditions have been derived for which the nonautonomous system exhibits globally attractive positive periodic solution. The nonautonomous system is shown to produce chaotic dynamics under the circumstances of the corresponding autonomous system that exhibits limit cycle oscillations. Overall, our findings deduce that media advertisements play a critical role in raising public awareness and ultimately resulting in the eradication of the disease from the society.