Üç avcı tek av modelinin caputo kesirli türevi ve geri besleme kontrol değişkeni ile analizi
Analysis of three predator-one prey model with caputo fractional derivative and feedback control
Yazar: SARE SAĞLAM
Danışman: PROF. DR. ÖMER FARUK GÖZÜKIZIL
Sakarya Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Ana Bilim Dalı / Matematik Bilim Dalı
Derece: Yüksek Lisans, 2024
Konu: Matematik = Mathematics

Özet:
Doğal yaşamın korunması ve sürdürülebilirliği açısından büyük öneme sahip av-avcı ilişkileri, ekosistemlerdeki popülasyon dengesini korur. Bu ilişkilerin ayrıntılı durumlarının varlığını belirlemek için av-avcı modeli ortaya konmuştur. Av-Avcı modeli, ekolojide ve biyolojide popülasyon dinamiklerini anlamak için kullanılan bir matematiksel modeldir. Bu modelde, bir av türünün (örneğin, geyikler) bir veya daha fazla avcı türü (örneğin, kurtlar) tarafından avlanması ve bu etkileşimin sonucunda her iki türün popülasyonlarının nasıl değiştiği incelenir. Bu tezde üç avcının bir av üzerindeki etkisini araştıran bir ekolojik model, genelleştirilmiş bir işlevsel tepki fonksiyonu ile incelenmiştir. Av popülasyonu için geri besleme kontrolünü içeren kesirli mertebeden üç avcı-tek av modeli üzerinde durulmuştur. Çevre ile avcı-av etkileşimindeki çeşitliliği modellemek amacıyla kapsamlı bir genelleştirilmiş fonksiyonel etkileşim sınıfı kullanılmıştır. Bu etkileşimler, çevre ve dört türün adaptasyonu gibi birçok faktörden etkilenebilir. Farklı denge noktalarının varlığı analiz edilmiş ve bu denge noktalarının asimptotik kararlılığını sağlamak için bazı yöntemler belirlenmiştir. Bu çalışmanın temel amacı, üç avcının bir av üzerindeki etkisini anlamak ve bu etkileşimlerin ekosistemdeki dengeyi nasıl etkilediğini belirlemektir. Beş farklı denge noktasının varlığı analiz edilmiştir. Bunlar, dört türün neslinin tükenme noktası, birinci avcının olmadığı denge, ikinci avcının olmadığı denge, üçüncü avcının olmadığı denge ve dört türün bir arada yaşayabildiği denge noktaları olduğu görülmüştür. Elde edilen bu noktaların çeşitli senaryolar oluşturabileceği  belirlenmiştir. Bu senaryolar arasında, dört popülasyonun neslinin tükenmesi, üç tür avcının neslinin tükenmesi, sırayla her bir yırtıcı popülasyonunun neslinin tükenmesi ve dört popülasyonun bir arada yaşaması bulunmaktadır. Teorik bulgular, , av ve yırtıcı türlerin birlikte yaşamasını yönetmedeki kritik önemini ve üç avcı ile tek bir avın bir arada yaşayabileceği olumlu denge için gerekli koşulların sağlandığını geri besleme kontrolü ile göstermektedir

Abstract:
Prey-predator relationships, which are of great importance for the protection and sustainability of natural life, maintain the population balance in ecosystems. The predator-prey model was introduced to determine the existence of detailed situations of these relationships.The Prey-Prey model is a mathematical model used in ecology and biology to understand population dynamics. This model examines the predation of a prey species (e.g., deer) by one or more predator species (e.g., wolves) and how the populations of both species change as a result of this interaction. The prey-predator model is a system in ecology that mathematically models the interaction between prey and predator species in an ecosystem. The predator-prey models a dynamic relationship that can be applied to many different areas, from natural life to human behavior, economics, and social systems.This relationship forms the basis of processes such as competition, seizing opportunities, avoiding dangers and establishing balance. The balance between prey and predator in nature provides a powerful metaphor for understanding many processesin human life.May vary depending on the fıeld purpose of use of the model.In general, artifıcial intelligence and machine learning models provide great advantages, especially in classifıcation and prediction tasks and offer signifıcant contributions in various sectors.This model is based on predators hunting prey species and these predators controlling the prey population. Prey-prey models are often described by equations that include the predation rate of predators on prey species and the reproduction rate of prey species. Such models are used to understand the dynamics of prey and predator populations in ecosystems and to predict future changes.Fractional derivative is a generalization of the concept of derivative beyond traditional integer order derivatives to fractional degrees.Mathematically, in classical differentiation, when taking the n-th derivative of a function,   n is an integer; in fractional differentiation,   n can be any real or complex number.Fractional derivatives have the ability to account for the effects of past information and events on the present. The Caputo derivative allows for modeling this effect in a more natural way because it takes into acconunt the memory properties of the system over time . This can be important in physical processes, biological systems, and fınancial modeling.Also, this model can be created using the Caputo fractional derivative. The Caputo fractional derivative is a derivative operator extended to non-integer orders of classical derivatives.The Caputo derivative acts as a bridge between classical and fractional derivatives and has applications particularly in fields such as engineering, physics, biology and economics.Such derivatives can be used to account for time lag and memory effects, allowing more accurate modeling of dynamic systems such as ecological systems. When the prey-prey model is created with Caputo fractional derivatives, the differential equations describing the changes in prey and predator populations over time are expressed with Caputo fractional derivatives. These differential equations may include the rate at which predators control the prey population, the rate at which predators hunt, and the rate at which the prey population reproduces. Caputo fractional derivatives are used to more precisely model the effects of derivatives with non-integer orders, unlike traditional differential equations. Integration of Caputo fractional derivatives into ecological modeling can help to more accurately account for complex interactions and environmental factors. Understanding Ecological Balance helps us understand how ecological balance is achieved and maintained by mathematically modeling the predator-prey relationships seen in natural life. In this way, appropriate conservation and management strategies can be developed to ensure that ecosystems function healthily and remain in balance. Study of Population Dynamics is used to understand how prey and predator populations change over time and how they affect each other. In this way, it is possible to understand the dynamics of populations in a particular ecosystem, such as growth, decline, competition and adaptation. Conservation of Natural Resources is important to understand and maintain the balance between prey and predator populations. Such models need to be used to sustainably manage natural resources and control hunting. Monitoring Changes in Ecosystems can be used to monitor and evaluate the effects of environmental changes and human intervention on prey and predator populations. In this way, the causes and consequences of changes in ecosystems can be better understood.In this paper, an ecological model with three predators competing over a prey with a generalized functional response function is examined. The effect of three predators on one prey was investigated by focusing on a fractional order three-predator-one-prey model that includes feedback control for the prey population. The reason for considering a comprehensive class of generalized functional interactions is to model diversity in predator-prey interactions with the environment. These interactions can be affected by many factors, such as the environment and the adaptation of the four species. By analyzing the existence of different equilibrium points, some situations have been derived to ensure the asymptotic stability of these equilibrium points. In our study, the three-predator-one-prey model was investigated with the Caputo fractional derivative using the feedback control variable. First, the variables of the equation system created with the three-predator-one-prey model were named. In the next steps, the existence of balance points were determined, respectively. The first balance point; equilibrium at which four species disappear, the second equilibrium point; balance free from predators, third balance point; The balance where the first and second hunters do not exist but the third hunter and prey exist, the fourth balance point; The balance where the first and third hunters do not exist but the second hunter and prey exist, the fifth balance point; It has been shown that the equilibrium point where the first predator and prey exist but not the second and third predators, and the sixth equilibrium point is the positive balance point where four species exist. Using these obtained balance points, the type and state of stability was examined. Finally, the opinion put forward is that when the necessary conditions are created, the four species become one. It has been stated that the positive balance resulting from living together will be stable. In this research, three predator and one prey were studied based on the study of the two predator and one prey model with the Caputo fractional derivative.The research can be expanded and focused on different numbers of prey and predator models, and the results are open to examination. The three predator one prey model allow exploring different ways to aproach a problem, both competitively an collaboratively.Each actor's perspective plays a significant role in the process of achieving the goal and ultimately success depends on the harmony of strategies and approacs.When we look at it, the general conclusion we reach is that the prey- predator model can provide signifıcant benefıts focus on improving the quality of life, using resources more effıciently, improving safety and health, producing environmentally friendly solutions and making various industries more effective. The correct and ethical use of the model effects for humanity in the future.