In the competition model for two species with populations N1 and N2 N2 b12 K1 dN1 N1 dt K1 dN2 N1 = 12N2 (1 – b21 dt K2 where only one species, N1, has limited carrying capacity. Nondimensionalise the system and determine the steady states. Investigate their stability and sketch the phase plane trajectories. Show that irrespective of the size of the parameters the principle of competitive exclusion holds. Briefly describe under what ecological cir- cumstances the species N2 becomes extinct.

a. Find the most general real-valued solution to the linear -2 -25 x. 1 -2 system of differential equations ' = x₁(t) x₂(t) = C1 b. In the phase plane, this system is best described as a source/ unstable node sink / stable node saddle center point / ellipses spiral source spiral sink none of these

You are given the following inhomogeneous system of first-order differentialequations for x(t) and y(t) in matrix form: x ̇ = 2x + y + 3 et ,y ̇ = 4x − y  What is the long-term behaviour of this particular solution as t becomes large? Does the ratio y/x tend to a fixed number, and if so what number?

Consider the system = (41%) (22) + (1) * น in which a is a constant (a) Determine the condition under which the system is controllable (b) For a 1 (i) Show that et4 = (cos(t)) sin(t)) cos(t)) (Hint: You may note that A4 = 1, A4k+1 = A, A4k+2 = −I, A4+3 = -A for all k > 0 and determine the MacLaurin series expansion of cos(t) and sin(t)) To (ii) Write the integral formula for the solution X (t) in terms of X (0) = X₁ = and u. Yo (ii) Extract a separate formula for each component of X(t) = ((0)