Predator prey interactions

Predator-prey interactions: 

Predator-prey interactions The Italian mathematician Vito Volterra (1860-1940) developed a model of predator-prey interaction in response to the data from Italian sea port Fiume (see the table).

Influencing factors: 

Influencing factors Natural births and deaths of sharks and fish in isolation from each other Decline the of fish population due to the fish being the prey of sharks Increase in the shark population due to the presence of more fish Fishing of both sharks and fish

Slide3: 

change in fish population = (fish born in isolation) - (fish deaths in isolation) – (fish eaten by sharks) – (fish caught by fishermen) change in shark population = (sharks born in isolation) – ( sharks deaths in isolation) + (extra sharks surviving due to fish) – (sharks caught by fishermen)

Modeling assumptions: 

Modeling assumptions The change in shark and fish populations, in isolation, is proportional to the present population of sharks and fish, respectively. The number of sharks and fish is directly proportional to the present populations, respectively. The number of fish eaten by sharks is directly proportional to the product of the number of fish present and the number of sharks present. The additional number of sharks surviving is directly proportional to the number of fish eaten.

Discrete version of Lotka-Volterra equations : 

Discrete version of Lotka-Volterra equations F(n) = number of fish after n time steps S(n) = number of sharks after n time steps F(n+1) – F(n) = change in fish population S(n+1) – S(n) = change in shark population

Discrete version of Lotka-Volterra equations: 

Discrete version of Lotka-Volterra equations F(n+1) – F(n) = aF(n) – bF(n) – kF(n)S(n) – rF(n) S(n+1) – S(n) = cS(n) – dS(n) + mF(n)S(n) – tS(n)

Discrete version of Lotka-Volterra equations: 

Discrete version of Lotka-Volterra equations F(n+1) – F(n) = (a-b-r)F(n)) – kF(n)S(n)) S(n+1) – S(n) = (c-d-t)S(n) + mF(n)S(n)

Steady- state solutions: 

Steady- state solutions 0 = (a-b-r)F(n) – kF(n)S(n) 0 = (c-d-t)S(n) + mF(n)S(n) Factorizing 0 = F(n)[(a-b-r) – kS(n))] 0 = S(n)[(c-d-t) + mF(n)] Either F(n) = 0 and S(n) = 0 Or F(n) = - (c-d-t)/m and S(n) = (a-b-r)/k

Steady- state solutions: 

Steady- state solutions 1. F(n) = 0 and S(n) = 0 means extinction of both species. 2. F(n) = - (c-d-t)/m and S(n) = (a-b-r)/k Steady fish population depends on c-d-t (which is shark growth) rate and m Steady shark population depends on a-b-r (which is fish growth rate) and k

Effect of fishing: 

Effect of fishing Consider less fishing circumstance (as in the First World War) F(n) = - (c-d-t)/m and S(n) = (a-b-r)/k So t and r decrease and all other parameters are constant

Effect of fishing: 

Effect of fishing F(n) = - (c-d-t)/m and S(n) = (a-b-r)/k If t and r decrease and all other parameters are constant the fish population decreases and the shark population increases The ratio sharks to fish increases

Question.: 

Question. An orchard is infested by a population of aphids. The aphids are preyed upon a type of beetle. The owner of the orchard decides to use a pesticide which kills a fixed fraction of both aphid and beetle. Use the steady-state solution of “Lotka-Volterra” equation to decide whether or not this is a wise move.

Predator-Prey: 

Predator-Prey a(n+1) = α·a(n) + β·[a(n)b(n)] (predator) b(n+1) = γ·b(n) − δ·[a(n)b(n)] (prey)

Competitive Hunter: 

Competitive Hunter a(n+1) = α·a(n) − β·[a(n)b(n)] b(n+1) = γ·b(n) − δ·[a(n)b(n)]

Mutualism: 

Mutualism a(n+1) = α·a(n) + [a(n)b(n)] b(n+1) = γ·b(n) + [a(n)b(n)]