Before we jump right in to the depth of the enticing world of **Prey Predator Mathematical Models**, it’s pertinent to establish a fine understanding of the fundamental concepts.

**Introduction to Prey Predator Mathematical Models**

The field of **mathematical biology** has manifested remarkable applications, and one such exquisite application is the **Prey Predator Mathematical Model**. This model, also known as the Lotka-Volterra equations, centers on the interaction dynamics between two species – the prey and the predator. These equations stand as a testament to the ever-compelling question — can the delicate balance of our ecosystem be quantified and predicted?

**Theoretical Framework of the Prey Predator Model**

The core foundation of the **Prey Predator Mathematical Model** is grounded in the systematic independent growth of both the prey and the predator populations. The model focuses on the symbiotic relationship, which is reciprocal and direct for these two species and their environment.

In the mathematical depiction of this framework, the rate of change in the population of the prey and the predator are denoted by ‘dx/dt’ and ‘dy/dt’ respectively. These rates are the product of their respective growth rates and population sizes, thus underlining the population’s intrinsic growth rate.

**Prey Population Dynamics**

With the absence of predators, the prey population would witness exponential growth. This surge in population aligns with the ‘law of unrestricted growth.’ Despite this growth, the population is kept in check by the existence of predators. This is captured by incorporating a term into the equation that signifies the interaction between the two species.

**Predator Population Dynamics**

On the other hand, the predator population sees a steady increase considering the availability of ample prey. The predator’s population declines as the prey’s population decreases, capturing the stark reality of a food chain’s hierarchy. The mathematical equation representative of this predator-prey dynamic is an amalgamation of the predator and prey interaction term as well as the intrinsic population decay rate of the predator, which showcases natural death rate and migration effects.

**Graphical Representation & Behavior**

The intriguing dynamic between preys and predators can be visualistically represented through phase plane diagrams also known as phase portraits. They underscore the state of the system over time, demonstrating an oscillatory behavior. The oscillations signify the population peaks and troughs of both species, underlining the dynamic equilibrium in their relationship.

**Factors Affecting Prey Predator Mathematical Models**

The **Prey Predator Mathematical Models** are impacted by various factors apart from the intrinsic growth rates of the species. These factors can include spatial distribution, delay in reproduction, disease outbreaks, and more. The understanding gained from these factors can aid in defining effective management strategies for conserving biodiversity and ensuring ecosystem balance.

**Limitations and Further Scope**

While highly insightful, the models also have limitations. They assume interactions are exclusively between two species, ignoring effectors such as habitat, food resources, and immigration. Current and future researchers aim to refine these models, exploring the effects of more complex multi-species interactions.

**Applications: Ecological Conservation & Beyond**

The **Prey Predator Mathematical Models** have far-reaching implications not just within ecology but also in fields such as economics, computer science, and even social behavior studies. They assist in fishery and wildlife management, pest control, and market trend predictions, highlighting their multifaceted versatility.

**Closing Thoughts**

In conclusion, the **Prey Predator Mathematical Models** are a riveting exploration into the natural balance of our ecosystems. Through their predictive ability and wide applicability, these mathematical models serve as compelling tools that bridge the gap between theoretical mathematics and practical world applications.

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