MATH3506 Mathematical Ecology Problem Sheet 3
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MATH3506 Mathematical Ecology Problem Sheet 3
1. The dynamics of 2 species is given by the system
= r1 N1 ╱ 1 一 
一 b12 
、
= r2 N2 ╱ 1 一 b21 
、
where b12 ≥ 0 and b21 ≥ 0. What kind of interactive behaviour is implied by the model? Scale the equations appropriately, analyse the stability of the steady states and plot the possible phase portraits. Under what circumstances does N2 become extinct? Comment on your results in an ecological context. (Based on Ex 3.1 in Murray, Mathematical Biology, Springer 1990.)
2. Consider the following model for 2 species cooperation:
N˙1 = r1 N1 (1 一 
+ a12 N2 ), N˙2 = r2 N2 (1 一 
+ a21 N1 )
where a12 , a21 > 0. Perform the linear stability analysis of each of the steady states when (a) there is an interior steady state and (b) there is no interior steady state.
3. The dynamics of 2 species is given by the system
dN1
dt
dN2
dt
= r1 N1 ╱ 1 一 
、
= r2 N2 ╱ 1 一 
、
where b12 ≥ 0 and b21 ≥ 0. What kind of interactive behaviour is implied by the model? Scale the equations appropriately and plot the phase portraits, being careful to dis- tinguish between topologically different phase portraits for different parameter values. What different behaviours can we expect? (Based on Ex 3.2 in Murray, Mathematical Biology, Springer 1990.)
2023-04-29