In the predator-prey model given:

$\frac{dx}{dt} = 3x - \frac{3x^2}{N} - 1.5xy$ $\frac{dy}{dt} = -2y + 3xy$

where $x$ represents the prey population and $y$ represents the predator population, the system describes interactions between the prey and predator populations. The terms in the equations explain how these populations evolve over time.

Breakdown of the equations:

1. Prey population ($x$) equation:

   • $3x$: The prey population grows at a rate proportional to its current size, indicating natural growth.
   • $- \frac{3x^2}{N}$: There is a limiting factor for the prey population, possibly due to competition for resources (such as food or space). This term represents a density-dependent limiting factor, unrelated to predators.
   • $-1.5xy$: This term models predation, where the predator population $y$ reduces the prey population $x$.

2. Predator population ($y$) equation:

   • $-2y$: The predator population naturally decreases without prey, likely due to lack of food.
   • $+3xy$: This term represents the growth of the predator population, which depends on the availability of prey.

Conclusion:

The prey population is being influenced by a limiting factor unrelated to predation, as seen in the $-\frac{3x^2}{N}$ term. This term shows that prey is also limited by something other than the predators, such as resources or space. Hence, the correct interpretation is that "The prey population has an additional limiting factor other than the predator," which matches the checked option in your image.

Would you like a detailed explanation of any specific part of this model?

Related Questions:

1. What is the biological meaning of each term in the predator-prey equations?
2. How does the term $-\frac{3x^2}{N}$ affect the prey population when $N$ changes?
3. Can the system reach equilibrium? If so, under what conditions?
4. How does changing the coefficient $1.5$ in $-1.5xy$ affect the dynamics of the system?
5. How would the system behave if $y$ (predator population) were to suddenly drop to zero?

Tip:

To fully understand predator-prey models, try visualizing the system using phase plane diagrams or solving for equilibrium points. This helps in understanding how populations interact over time.