The predator-prey model MTH 347 Project ✓ Solved

The predator-prey model investigates the non-linear, predator-prey system described by x′ = 0.06x − 0.0004yx and y′ = −0.08y + 0.0002xy. This system models the populations of rabbits (x) and foxes (y) in a hypothetical nature preserve, where time (t) is measured in months. This report will provide insights into the dynamics of these populations, represented through graphical analysis, looking to fulfill the assigned tasks effectively.

Introduction

Understanding the dynamics of biological systems aids in wildlife management and conservation efforts. The predator-prey model is crucial in representing how two species interact within an ecosystem. Predators (foxes) and prey (rabbits) exhibit fluctuating population dynamics influenced by their interactions. This project employs MATLAB to visualize these interactions through phase portraits and to analyze population behaviors over time.

Setting Up the Model in MATLAB

The first step involves running the MATLAB program pplane8 to generate the direction field and phase portrait of the system. By selecting 'predator prey' in the gallery menu, we can explore how the populations of rabbits and foxes change with various parameters. It is important to adjust the minimum and maximum population values to visualize equilibrium points effectively.

Direction Field and Phase Portrait

Once the system is set up in MATLAB, the direction field and phase portrait can be generated. The direction arrows indicate the flow of populations over time; they show the trajectories that populations would follow given initial conditions. The equilibrium point can be identified visually where trajectories converge, representing a stable population level where births and deaths balance out. In this model, we identify the predators (foxes) and prey (rabbits) by observing the direction of the arrows; the population that increases when the other decreases is the prey. Thus, rabbits are prey, and foxes are the predators.

Direction Field and Phase Portrait

Population Dynamics Over Time

In Step 2, we assume an initial condition of 200 rabbits and 50 foxes. Generating a plot of both populations over time indicates oscillatory behavior typical of predator-prey interactions. The rabbit population is expected to reach its maximum before the fox population due to the nature of the relationship. Through observation, it's determined that the maximum rabbit population occurs at approximately t = 10 months to t = 15 months, while the fox population peaks subsequent to this, suggesting a delay in their response to the increasing food supply.

Rabbit and Fox Populations Over Time

Estimating Minimums and Maximums

Close evaluation reveals that the rabbit population fluctuates between 180 - 250 individuals, whereas the fox population oscillates between 30 - 70 individuals. The period of oscillation for both populations hovers around 25 months, indicating a cyclical pattern characteristic of their environmental interactions. This cyclical behavior suggests a balanced ecosystem, although extreme fluctuations can lead to instability. If, hypothetically, a disease wipes out the fox population, the model predicts a significant spike in the rabbit population, potentially leading to resource depletion and eventual population collapse.

Management Strategies for Population Stability

In Step 3, as a manager of the nature preserve, addressing the high fluctuations in populations is crucial. Observing the trajectories, it's advisable to remove foxes when their population approaches its peak (around t = 15 months). This would help stabilize the rabbit population before it crashes due to overpopulation. The removal of approximately one-third of the fox population (around 20 individuals) can reduce competition and allow prey populations to stabilize. However, if this reduction is executed too late or the number is misjudged, it could destabilize the ecosystem further, leading to overpopulation of rabbits and subsequent starvation.

Conclusion

This project illustrates the complexity of predator-prey relationships and the utility of mathematical modeling in ecological management. Through graphical analysis and dynamic simulations in MATLAB, we understand that careful observation and timely intervention are key to maintaining balance in nature preserves. Continued research and real-time data collection would further enhance our management strategies, fostering a sustainable coexistence between rabbits and foxes.

References

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