Problem Statement
A competing species biological system is described by the nonlinear autonomous differential equations:\n\n$$\frac{dx}{dt} = x(3 - x - 2y), \quad \frac{dy}{dt} = y(2 - x - y)$$
**(a)** [3 marks] Find all four equilibrium points in the first quadrant ($x \ge 0, y \ge 0$).
**(b)** [3 marks] Determine the general Jacobian matrix $\mathbf{J}(x, y)$ of the system.
**(c)** [4 marks] Evaluate the Jacobian matrix at the coexistence equilibrium point $(1, 1)$ and determine the stability and type of equilibrium.
Verified Solution & Marking Scheme
Part (a) [M1]
Equilibrium requires $\dot{x} = 0$ and $\dot{y} = 0$. Case 1: $x = 0, y = 0 \implies (0, 0)$
Part (a) [A1]
Case 2: $x = 0 \implies 2 - y = 0 \implies (0, 2)$. Case 3: $y = 0 \implies 3 - x = 0 \implies (3, 0)$
Part (a) [A1]
Case 4: $x + 2y = 3$ and $x + y = 2 \implies y = 1, x = 1$. Points are $(0, 0), (3, 0), (0, 2), (1, 1)$.
Part (b) [M1]
$f(x, y) = 3x - x^2 - 2xy \implies \frac{\partial f}{\partial x} = 3 - 2x - 2y, \quad \frac{\partial f}{\partial y} = -2x$
Part (b) [A1]
$g(x, y) = 2y - xy - y^2 \implies \frac{\partial g}{\partial x} = -y, \quad \frac{\partial g}{\partial y} = 2 - x - 2y$
Part (b) [A1]
$\mathbf{J}(x, y) = \begin{pmatrix} 3 - 2x - 2y & -2x \\ -y & 2 - x - 2y \end{pmatrix}$
Part (c) [M1]
At $(1, 1)$: $\mathbf{J}(1, 1) = \begin{pmatrix} 3 - 2 - 2 & -2 \\ -1 & 2 - 1 - 2 \end{pmatrix} = \begin{pmatrix} -1 & -2 \\ -1 & -1 \end{pmatrix}$
Part (c) [A1]
Characteristic eq: $\det(\mathbf{J} - \lambda\mathbf{I}) = (-1-\lambda)^2 - 2 = \lambda^2 + 2\lambda - 1 = 0$
Part (c) [A1]
$\lambda = \frac{-2 \pm \sqrt{4 + 4}}{2} = -1 \pm \sqrt{2} \implies \lambda_1 = -1 + \sqrt{2} > 0, \quad \lambda_2 = -1 - \sqrt{2} < 0$
Part (c) [R1]
Since eigenvalues have opposite real signs, $(1, 1)$ is an unstable saddle point.