Dynamical Analysis of the Symbiotic Model of Commensalism in Four Populations with Michaelis-Menten type Harvesting in the First Commensal Population

This study discusses the dynamical analysis of the symbiosis commensalism and parasitism models in four populations with Michaelis-Menten type harvesting in the first commensal population. This model is formed from a construction of the symbiotic model of commensalism and parasitism by harvesting the commensal population. This construction is by adding a new population, namely the second commensal population. Furthermore, it will be investigated that the four populations can coexist. The first analysis is to identify the conditions of existence at all equilibrium points along with the conditions for their existence and local stability around the equilibrium point along with the stability requirements. From this model, it is obtained sixteen points of equilibrium, namely one point of extinction in the four populations, four points of extinction in all three populations, six points of extinction in both populations, four points of extinction in one population and one point where the four populations can coexist. Of the sixteen points, only four points can be asymptotically stable if they meet the stability conditions that have been determined. Finally, a numerical simulation is performed to describe the model behavior. In this study, the method used in numerical simulation is the RK-4 method. The numerical simulation results that have been obtained support the dynamical analysis results that have been carried out previously

Dynamics of a harvested cyanobacteria-fish model with modified Holling type Ⅳ functional response

In this paper, considering the aggregation effect and Allee effect of cyanobacteria populations and the harvesting of both cyanobacteria and fish by human beings, a new cyanobacteria-fish model with two harvesting terms and a modified Holling type Ⅳ functional response function is proposed. The main purpose of this paper is to further elucidate the influence of harvesting terms on the dynamic behavior of a cyanobacteria-fish model. Critical conditions for the existence and stability of several interior equilibria are given. The economic equilibria and the maximum sustainable total yield problem are also studied. The model exhibits several bifurcations, such as transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. It is concluded from a biological perspective that the survival mode of cyanobacteria and fish can be determined by the harvesting terms. Finally, concrete examples of our model are given through numerical simulations to verify and enrich the theoretical results

Two predators one prey model that integrates the effect of supplementary food resources due to one predator's kleptoparasitism under the possibility of retribution by the other predator

In ecology, foraging requires animals to expend energy in order to obtain resources. The cost of foraging can be reduced through kleptoparasitism, the theft of a resource that another individual has expended effort to acquire. Thus, kleptoparasitism is one of the most significant feeding techniques in ecology. In this study, we investigate a two predator one prey paradigm in which one predator acts as a kleptoparasite and the other as a host. This research considers the post-kleptoparasitism scenario, which has received little attention in the literature. Parametric requirements for the existence as well as local and global stability of biologically viable equilibria have been proposed. The occurrences of various one parametric bifurcations, such as saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation, as well as two parametric bifurcations, such as Bautin bifurcation, are explored in depth. Relatively low growth rate of first predator induces a subcritical Hopf bifurcation although a supercritical Hopf bifurcation occurs at relatively high growth rate of first predator making coexistence of all three species possible. Some numerical simulations have been provided for the purpose of verifying our theoretical conclusions

STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN ECOLOGY AND EPIDEMICS

Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and predictions in various areas of life sciences, such as population dynamics, epidemiology, immunology, physiology, and neural networks. The memory or time-delays, in these models, are related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on. In ordinary differential equations (ODEs), the unknown state and its derivatives are evaluated at the same time instant. In DDEs, however, the evolution of the system at a certain time instant depends on the past history/memory. Introduction of such time-delays in a differential model significantly improves the dynamics of the model and enriches the complexity of the system. Moreover, natural phenomena counter an environmental noise and usually do not follow deterministic laws strictly but oscillate randomly about some average values, so that the population density never attains a fixed value with the advancement of time. Accordingly, stochastic delay differential equations (SDDEs) models play a prominent role in many application areas including biology, epidemiology and population dynamics, mostly because they can offer a more sophisticated insight through physical phenomena than their deterministic counterparts do. The SDDEs can be regarded as a generalization of stochastic differential equations (SDEs) and DDEs.This dissertation, consists of eight Chapters, is concerned with qualitative and quantitative features of deterministic and stochastic delay differential equations with applications in ecology and epidemics. The local and global stabilities of the steady states and Hopf bifurcations with respect of interesting parameters of such models are investigated. The impact of incorporating time-delays and random noise in such class of differential equations for different types of predator-prey systems and infectious diseases is studied. Numerical simulations, using suitable and reliable numerical schemes, are provided to show the effectiveness of the obtained theoretical results.Chapter 1 provides a brief overview about the topic and shows significance of the study. Chapter 2, is devoted to investigate the qualitative behaviours (through local and global stability of the steady states) of DDEs with predator-prey systems in case of hunting cooperation on predators. Chapter 3 deals with the dynamics of DDEs, of multiple time-delays, of two-prey one-predator system, where the growth of both preys populations subject to Allee effects, with a direct competition between the two-prey species having a common predator. A Lyapunov functional is deducted to investigate the global stability of positive interior equilibrium. Chapter 4, studies the dynamics of stochastic DDEs for predator-prey system with hunting cooperation in predators. Existence and uniqueness of global positive solution and stochastically ultimate boundedness are investigated. Some sufficient conditions for persistence and extinction, using Lyapunov functional, are obtained. Chapter 5 is devoted to investigate Stochastic DDEs of three-species predator prey system with cooperation among prey species. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution to the model are established, by constructing a suitable Lyapunov functional. Chapter 6 deals with stochastic epidemic SIRC model with time-delay for spread of COVID-19 among population. The basic reproduction number ℛs0 for the stochastic model which is smaller than ℛ0 of the corresponding deterministic model is deduced. Sufficient conditions that guarantee the existence of a unique ergodic stationary distribution, using the stochastic Lyapunov functional, and conditions for the extinction of the disease are obtained. In Chapter 7, some numerical schemes for SDDEs are discussed. Convergence and consistency of such schemes are investigated. Chapter 8 summaries the main finding and future directions of research. The main findings, theoretically and numerically, show that time-delays and random noise have a significant impact in the dynamics of ecological and biological systems. They also have an important role in ecological balance and environmental stability of living organisms. A small scale of white noise can promote the survival of population; While large noises can lead to extinction of the population, this would not happen in the deterministic systems without noises. Also, white noise plays an important part in controlling the spread of the disease; When the white noise is relatively large, the infectious diseases will become extinct; Re-infection and periodic outbreaks can also occur due to the time-delay in the transmission terms

Repulsive-attractive models for the impact of two predators on prey species varying in anti-predator response

This study considers the dynamical interaction of two predatory carnivores (Lions (Panthera leo) and Spotted Hyaenas (Crocuta crocuta)) and three of their common prey (Buffalo (Syncerus caffer), Warthog (Phacochoerus africanus) and Kudu (Tragelaphus strepsiceros)). The dependence on spatial structure of species’ interaction stimulated the author to formulate reaction-diffusion models to explain the dynamics of predator-prey relationships in ecology. These models were used to predict and explain the effect of threshold populations, predator additional food and prey refuge on the general species’ dynamics. Vital parameters that model additional food to predators, prey refuge and population thresholds were given due attention in the analyses. The stability of a predator-prey model for an ecosystem faced with a prey out-flux which is analogous to and modelled as an Allee effect was investigated. The results highlight the bounds for the conversion efficiency of prey biomass to predator biomass (fertility gain) for which stability of the three species ecosystem model can be attained. Global stability analysis results showed that the prey (warthog) population density should exceed the sum of its carrying capacity and threshold value minus its equilibrium value i.e., W >(Kw + $) −W . This result shows that the warthog’s equilibrium population density is bounded above by population thresholds, i.e., W < (Kw+$). Besides showing the occurrence under parameter space of the so-called paradox of enrichment, early indicators of chaos can also be deduced. In addition, numerical results revealed stable oscillatory behaviour and stable spirals of the species as predator fertility rate, mortality rate and prey threshold were varied. The stabilising effect of prey refuge due to variations in predator fertility and proportion of prey in the refuge was studied. Formulation and analysis of a robust mathematical model for two predators having an overlapping dietary niche were also done. The Beddington-DeAngelis functional and numerical responses which are relevant in addressing the Principle of Competitive Exclusion as species interact were incorporated in the model. The stabilizing effect of additional food in relation to the relative diffusivity D, and wave number k, was investigated. Stability, dissipativity, permanence, persistence and periodicity of the model were studied using the routine and limit cycle perturbation methods. The periodic solutions (b 1 and b 3), which influence the dispersal rate (') of the interacting species, have been shown to be controlled by the wave number. For stability, and in order to overcome predator natural mortality, the nutritional value of predator additional food has been shown to be of high quality that can enhance predator fertility gain. The threshold relationships between various ecosystem parameters and the carrying capacity of the game park for the prey species were also deduced to ensure ecosystem persistence. Besides revealing irregular periodic travelling wave behaviour due to predator interference, numerical results also show oscillatory temporal dynamics resulting from additional food supplements combined with high predation rates

Mathematical Modeling of Biological Systems

Mathematical modeling is a powerful approach supporting the investigation of open problems in natural sciences, in particular physics, biology and medicine. Applied mathematics allows to translate the available information about real-world phenomena into mathematical objects and concepts. Mathematical models are useful descriptive tools that allow to gather the salient aspects of complex biological systems along with their fundamental governing laws, by elucidating the system behavior in time and space, also evidencing symmetry, or symmetry breaking, in geometry and morphology. Additionally, mathematical models are useful predictive tools able to reliably forecast the future system evolution or its response to specific inputs. More importantly, concerning biomedical systems, such models can even become prescriptive tools, allowing effective, sometimes optimal, intervention strategies for the treatment and control of pathological states to be planned. The application of mathematical physics, nonlinear analysis, systems and control theory to the study of biological and medical systems results in the formulation of new challenging problems for the scientific community. This Special Issue includes innovative contributions of experienced researchers in the field of mathematical modelling applied to biology and medicine

Dynamical Models of Biology and Medicine

Mathematical and computational modeling approaches in biological and medical research are experiencing rapid growth globally. This Special Issue Book intends to scratch the surface of this exciting phenomenon. The subject areas covered involve general mathematical methods and their applications in biology and medicine, with an emphasis on work related to mathematical and computational modeling of the complex dynamics observed in biological and medical research. Fourteen rigorously reviewed papers were included in this Special Issue. These papers cover several timely topics relating to classical population biology, fundamental biology, and modern medicine. While the authors of these papers dealt with very different modeling questions, they were all motivated by specific applications in biology and medicine and employed innovative mathematical and computational methods to study the complex dynamics of their models. We hope that these papers detail case studies that will inspire many additional mathematical modeling efforts in biology and medicin

Nonlinear dynamics of plankton ecosystem with impulsive control and environmental fluctuations

It is well known that the density of plankton populations always increases and decreases or keeps invariant for a long time, and the variation of plankton density is an important factor influencing the real aquatic environments, why do these situations occur? It is an interesting topic which has become the common interest for many researchers. As the basis of the food webs in oceans, lakes, and reservoirs, plankton plays a significant role in the material circulation and energy flow for real aquatic ecosystems that have a great effect on the economic and social values. Planktonic blooms can occur in some environments, however, and the direct or indirect adverse effects of planktonic blooms on real aquatic ecosystems, such as water quality, water landscape, aquaculture development, are sometimes catastrophic, and thus planktonic blooms have become a challenging and intractable problem worldwide in recent years. Therefore, to understand these effects so that some necessary measures can be taken, it is important and meaningful to investigate the dynamic growth mechanism of plankton and reveal the dynamics mechanisms of formation and disappearance of planktonic blooms. To this end, based on the background of the ecological environments in the subtropical lakes and reservoirs, this dissertation research takes mainly the planktonic algae as the research objective to model the mechanisms of plankton growth and evolution. In this dissertation, some theories related to population dynamics, impulsive control dynamics, stochastic dynamics, as well as the methods of dynamic modeling, dynamic analysis and experimental simulation, are applied to reveal the effects of some key biological factors on the dynamics mechanisms of the spatial-temporal distribution of plankton and the termination of planktonic blooms, and to predict the dynamics evolutionary processes of plankton growth. The main results are as follows: Firstly, to discuss the prevention and control strategies on planktonic blooms, an impulsive reaction-diffusion hybrid system was developed. On the one hand, the dynamic analysis showed that impulsive control can significantly influence the dynamics of the system, including the ultimate boundedness, extinction, permanence, and the existence and uniqueness of positive periodic solution of the system. On the other hand, some experimental simulations were preformed to reveal that impulsive control can lead to the extinction and permanence of population directly. More precisely, the prey and intermediate predator populations can coexist at any time and location of their inhabited domain, while the top predator population undergoes extinction when the impulsive control parameter exceeds some a critical value, which can provide some key arguments to control population survival by means of some reaction-diffusion impulsive hybrid systems in the real life. Additionally, a heterogeneous environment can affect the spatial distribution of plankton and change the temporal-spatial oscillation of plankton distribution. All results are expected to be helpful in the study of dynamic complex of ecosystems. Secondly, a stochastic phytoplankton-zooplankton system with toxic phytoplankton was proposed and the effects of environmental stochasticity and toxin-producing phytoplankton (TPP) on the dynamics mechanisms of the termination of planktonic blooms were discussed. The research illustrated that white noise can aggravate the stochastic oscillation of plankton density and a high-level intensity of white noise can accelerate the extinction of plankton and may be advantageous for the disappearance of harmful phytoplankton, which imply that the white noise can help control the biomass of plankton and provide a guide for the termination of planktonic blooms. Additionally, some experimental simulations were carried out to reveal that the increasing toxin liberation rate released by TPP can increase the survival chance of phytoplankton population and reduce the biomass of zooplankton population, but the combined effects of those two toxin liberation rates on the changes in plankton are stronger than that of controlling any one of the two TPP. All results suggest that both white noise and TPP can play an important role in controlling planktonic blooms. Thirdly, we established a stochastic phytoplankton-toxic producing phytoplankton-zooplankton system under regime switching and investigated how the white noise, regime switching and TPP affect the dynamics mechanisms of planktonic blooms. The dynamical analysis indicated that both white noise and toxins released by TPP are disadvantageous to the development of plankton and may increase the risk of plankton extinction. Also, a series of experimental simulations were carried out to verify the correctness of the dynamical analysis and further reveal the effects of the white noise, regime switching and TPP on the dynamics mechanisms of the termination of planktonic blooms. On the one hand, the numerical study revealed that the system can switch from one state to another due to regime shift, and further indicated that the regime switching can balance the different survival states of plankton density and decrease the risk of plankton extinction when the density of white noise are particularly weak. On the other hand, an increase in the toxin liberation rate can increase the survival chance of phytoplankton but reduce the biomass of zooplankton, which implies that the presence of toxic phytoplankton may have a positive effect on the termination of planktonic blooms. These results may provide some insightful understanding on the dynamics of phytoplankton-zooplankton systems in randomly disturbed aquatic environments. Finally, a stochastic non-autonomous phytoplankton-zooplankton system involving TPP and impulsive perturbations was studied, where the white noise, impulsive perturbations and TPP are incorporated into the system to simulate the natural aquatic ecological phenomena. The dynamical analysis revealed some key threshold conditions that ensure the existence and uniqueness of a global positive solution, plankton extinction and persistence in the mean. In particular, we determined if there is a positive periodic solution for the system when the toxin liberation rate reaches a critical value. Some experimental simulations also revealed that both white noise and impulsive control parameter can directly influence the plankton extinction and persistence in the mean. Significantly, enhancing the toxin liberation rate released by TPP increases the possibility of phytoplankton survival but reduces the zooplankton biomass. All these results can improve our understanding of the dynamics of complex of aquatic ecosystems in a fluctuating environment

Modelling the effects of odours and spying parasitoids on fruit fly population dynamics

The study of the role of chemical information in species interactions has been mostly restricted to studies at the level of individual organisms. The central question in this thesis is how intraspecific chemical information conveyance and exploitation thereof by a natural enemy affects the spatial population dynamics of a species. To answer this question, I developed a spatio-temporal model where both host and parasitoid can respond to infochemicals. Our model system consists of the fruit fly Drosophila melanogaster, and its natural enemy, the larval parasitoid Leptopilina heterotoma. D. melanogaster uses its aggregation pheromone in combination with odours from fermenting fruits to localise suitable resources for reproduction. L. heterotoma uses these same odours to localise its host. For D. melanogaster, aggregation on a resource can be beneficial when a population is small and has to overcome negative effects associated with low population densities. Such negative effects, known as the Allee effect, can for instance be caused by difficulties in resource exploitation or in finding a mate. Aggregation also involves costs. Individuals within an aggregation frequently experience more severe competition for food, space and mates than they would experience when being on their own. Furthermore, I investigated which behavioural decisions enhance the ability to find - and distinguish between – odour sources that differ in their suitability for reproduction. On the individual level, this research showed that, like real fruit flies, the modelled fruit flies need to have a preference for the presence of both aggregation pheromone and food odours, over food odours alone, to be able to distinguish between the two types of odour sources. The results show that this stronger preference does not have to be innate. As long as fruit flies are able to remember and adjust their current preference based on the odour concentrations that they perceive, more fruit flies find the more attractive odour source. On a population level, this thesis shows that the use of chemical information by D. melanogaster affects its population dynamics. In the absence of its natural enemy, and when the Drosphila population is small, the use of food odours and aggregation pheromone has a positive effect on population growth and enhances the fruit fly’s colonization ability. When the population becomes larger, however, the negative effects of larval competition are stronger than the positive effects of reduced mortality due to the Allee effect. The use of chemical information was crucial to colonize an area from the boundaries. A fruit fly population that was unable to use chemical information could not colonize the area and went extinct. When parasitoids can use chemical information, parasitism rates are higher, resulting in a slower population growth of their host. No difference was recorded in fruit fly population size and in larval mortality due to parasitism, when parasitoids exploited the aggregation pheromone of the fruit fly adults as compared with the simulations where the parasitoids could only respond to chemicals emitted by the host habitat. In contrast, the use of chemical information by the host enhanced its population growth and enabled it to survive, even at higher parasitoid densities. This research showed that mortality when the population was small had a greater impact on population size than mortality due to competition or parsitism. Food patches are not always abundant in nature. Thus, the reproductive success of fruit flies is mainly determined by their opportunities of producing clutches (i.e. locating patches) rather than by preventing over-aggregation or parasitism. As a result, the use of chemical information has a net positive effect on fruit fly population dynamics, despite the fact that L. heterotoma is able to exploit it.</p