Final answer:
For a predation parameter p of 0.5, both the predator and the prey populations modeled by the matrix will eventually die out. A value of p = 0.7 corresponds to a system where both populations remain constant, with their relative sizes determined by the corresponding eigenvector.
Step-by-step explanation:
The question involves analyzing a predator-prey matrix to determine the fate of the populations based on different values of the predation parameter p.
When the predation parameter p is set to 0.5, we can insert it into the matrix A resulting in A = (0.4 0.3) (-0.5 1.2). The long-term behavior of the system can be examined by looking at the eigenvalues of this matrix. If both eigenvalues are less than one in absolute value, both populations will eventually die out. In this case, calculating the eigenvalues show that indeed both eigenvalues are less than one, confirming the eventual extinction of both populations.
To find the value of p that keeps both populations constant, we would need to determine the value that makes the leading eigenvalue of the matrix exactly one, indicating a steady state. Solving the characteristic equation for the matrix with an eigenvalue of one, we find that the value of p that fulfills this condition is 0.7.
Furthermore, by analyzing the eigenvector associated with the eigenvalue of one, we can find the relative population sizes in this steady-state scenario. The calculations show that the relative population sizes are not equal, due to the differing impacts of the predator and prey in the matrix.