Question

Exercise 3, Section 9.5. Modified Lotka- Volterra Predator-Prey model Consider two species (rabbits and foxes) such that the population R (rabbits) and F (foxrs) obey the system of equations dR dt dF dt R2-R)-12RF . What happens to the population of rabbits if the number of foxes is arro? (Use the phase line analysis from Chapter 2) What happens to the population of foxes if the number of rabbits is zero? 3. Using the method of nullclines, draw an approximate phase portrait and find the equilibrium Determine the nature of each critical point by analyzing the corresponding linear system . A terrible illness spreads among the foxes, decreasing the fox population growth rate such points of the system What do you conclude about the long-term survival prospects of each species? that it obeys the modified equation dt How does this change the long-term balance of the ecosystem

Answer 1

dt dヒ 尺 dヒ Then ,r~. .. Rabbit populdam achieved steady stateR) fter arge time t rabbits - o () 9 no of dt becomes 0 Then for porulaien decrase aud ally b

B) Nullines dl t 0,160) (1,0.833) inc reases) dt 1 R-1 40 C lio) eureases dt R-10 (110.833 mi R-R-120 0,0) の at 2-140 Resnltant

equilibrium paints ave intrsedion of nullcline and f nultines W) linear Analysis: (ompuxte丁ALobian ,丁 ITE tF gf -I+R determinant= -2<0 00 saddle node

T-2-24] at (210) T- T <o rate oo Saddle node 049-1, 2.1 Trace =-049 0.333 Aeterminant (d)b.94 o stable mode
​​​​​​in long term, both species survive.