Chapter 6: Problem 2
Assume the given autonomous system models the population dynamics of two species, \(x\) and \(y\), within a colony. (a) For each of the two species, answer the following questions. (i) In the absence of the other species, does the remaining population continuously grow, decline toward extinction, or approach a nonzero equilibrium value as time evolves? (ii) Is the presence of the other species in the colony beneficial, harmful, or a matter of indifference? (b) Determine all equilibrium points lying in the first quadrant of the phase plane (including any lying on the coordinate axes). (c) The given system is an almost linear system at the equilibrium point \((x, y)=(0,0)\). Determine the stability properties of the system at \((0,0)\). $$ \begin{aligned} &x^{\prime}=-x-x^{2} \\ &y^{\prime}=-y+x y \end{aligned} $$
Short Answer
Step by step solution
Key Concepts
These are the key concepts you need to understand to accurately answer the question.