Dynamic Analysis Of Two Biological Filippov Systems With Threshold Policy Control

The world now faces a global extinction crisis never witnessed by humankind.Scientists predict that more than one million species are on track for extinction in the coming decades.The current extinction crisis is one of the most severe threats to the ecosystems of the world in and it's the entire of people making.The destruction,pollution,the spread of invasive species without studied,climate change,the high population,and other human activities,and in particular,overharvesting of natural populations and fragmentation of natural habitats have pushed nature to the brink.In light of the current global mass extinction of species,ecologists are facing great challenges addressing the extinction crisis.Because every species'extinction potentially leads to the extinction of others bound to that species in the complex food chains,numbers of extinctions are likely to snowball in the coming decades as ecosystems unravel.Therefore the ecologists began to discuss and study the extinction.In past years,scientists used smooth differential equations to describe ecology systems,but recently there are some applications that are better designed by using discontinuous ordinary differential equations which are called Filippov systems.The Filippov system provides a natural and convenient unified framework for the mathematical modeling of several real-world problems.The Filippov system can be described in its simplest modelization,in which the discontinuity surface splits the state space into two regions.When the solution trajectory stays inside the same region and does not reach the discontinuity surface,then the dynamics behave as in a conventional smooth system.But when the trajectory reaches the discontinuity surface there are two possible outcomes that can be occurred,either the trajectory crosses it,or the trajectory stays on it.In the last case,a description of the motion on the discontinuity surface will be required to describe a sliding motion.Consequently,Filippov systems can be used to represent many biological models.These phenomena described as an instantaneous event interrupts the smooth evolution of the processes in which the evolution switches between two or more distinct states.Thus,a different set of differential equations or maps can be used for describing each state when these states are modeled.Therefore,in order to avoid high extinction risks of prey,middle predator,and top predator,and keep the stability of the three species food chain model,in the first part of this thesis,we introduce two different models of three species food chain models.Firstly,in Chapter 2,we introduce a Filippov food chain model under threshold policy control.To describe the three species food chain interaction model composed of prey,middle predator,and top predator.The middle predator preys on prey and the top predator preys on the middle predator.The threshold policy is designed to play a pivotal strategy for controlling the three species in the Filippov food chain model.In this strategy,the control techniques of the exploited natural resources are used to modulate the harvesting effort to avoid high risks of extinction of the middle predator and keep the stability of the food chain,by prohibiting fishing when the population density drops below a prescribed threshold.Filippov systems,have been shown to be effective in descript the food chain sy stems with threshold policy control.where threshold policies very useful in managing renewable resources,being simple to design and implement,and also yielding advantages in situations of overexploitation.The dynamic behavior of the system stability of the regular and virtual equilibria are discussed.Also,the dy namic behavior of sliding mode and its equilibria(pseudo equilibria),is discussed.We have got stable pseudo equilibria which indicate that there is coexist between the system at the discontinuity surface.The complicated non-smooth dynamic behaviors(sliding and crossing segment and their domains)are analyzed.The sufficient conditions for the existence of the bifurcation set of pseudo-equilibrium and the grazing sliding bifurcation have been investigated.The group of special cases was studied with a change in the parameters to see how the threshold affected the stability of the system.It was found that the threshold plays an important role in the stability of the system and the coexistence of his species.Secondly,in Chapter 3,we introduce a Filippov food chain model with Holling ty pe II under threshold policy control.In this chapter,we used the control techniques of the exploited natural resources to modulate the harvesting effort to avoid high risks of extinction of the prey and keep the stability of the food chain.No control is applied if the density of the prey population is less than the threshold,then the exploitation is forbidden.Meanwhile,exploitation is permitted if the density of the prey population increases and exceeds the threshold.We have found that this system has complex dynamics including chaotic effects.The dynamic behaviors and the bifurcation sets of this model including the existence and stability of different types of equilibria are discussed analytically and numerically.Moreover,the regions of the sliding and crossing segments are analyzed.The dy namic behaviors of sliding mode including the bifurcation sets of pseudo-equilibria and the sliding bifurcations(crossing sliding and grazing sliding bifurcation)are investigated.Numerically,the bifurcation diagram and maximum Lyapunov exponents are carried out to show the complex dynamics of Filippov food chain model,for instance,it has stable periodic,double periodic,and chaotic solutions as well as double periodic sliding bifurcation.It has been shown that the threshold policy control can be easily implemented and used for stabilizing the chaotic behavior of Filippov food chain model.Finally,the quality and safety of crops are of paramount importance to humans where the crops form an essential part of the diet of humans yet.Almost all of the carbohydrate we consume is derived from food crops,either directly or as the result of processing and extraction.During crop production,harvesting,postharvest handling,and storage they can be affected by pests with the potential to cause foodborne illness.Consequently,the threat posed to crop production by plant pests and diseases is one of the key factors that could lead to instability in global food security,and the problem is forecast to get worse,scientists warn.Whereas the most critical factor for the increase in crop production is the successful resistance of pests and pathogens to stability in global food security.In recent years,people utilized many pesticides to hold pest down to improve the yield of crops,however,the unreasonable utilization of pesticides will make the pests drugresistant,contaminate the environment,and furthermore affects the life of people and animals.So how to decrease the effect brought by pests but doing no damage to the living condition of people and animals is an issue which has attracted the great concern of ecologists and biomathematics specialists.Integrated Pest Management(IPM)is an ecosystem approach to crop production and protection that combines different management strategies and practices to grow healthy crops and minimize the use of pesticides.IPM is an approach-based method for analysis of the agro-ecosystem and the management of its different elements to control pest and keep them at an acceptable level(threshold)with respect to the economic,health and environmental.In order to ensure the realism of the predator-prey models and illustrate how the population dynamics of these models depend on the past relevant information,it is important to integrate delays into these models.The time delay in prey-predator models is due to two reasons.The first one is the time of gestation,and the other is the time of maturation.Thus time delays significantly impact the overall properties of predator-prey models.With this motivation,in the second part of this thesis,in Chapter 4,a Filippov prey-predator(pest-natural enemy)model with time delay is introduced to describe pest control models,where the time delay represents the change of growth rate of the natural enemy before releasing it to feed on pests.The threshold conditions for the stability of the equilibria are derived by using the time delay as a bifurcation parameter.It is shown that when the time delay parameter passes through some critical values,a periodic oscillation phenomenon appears through Hopf bifurcation.Further,by using Filippov's convex method we obtain the equation of sliding motion and analysis the sliding mode dynamics.Numerically,we demonstrate that the time delay plays a substantial role in discontinuity-induced bifurcation.More precisely,one can get boundary focus bifurcation from boundary node bifurcation through variation of the value of the time delay.Moreover,the time delay is used as a bifurcation parameter to obtain sliding-switching and sliding-grazing bifurcations.In conclusion,a Filippov system with time delay can give new insights into pest control models.The analytical findings of this thesis are verified through numerical investigations.