MA 138: Calculus II for the Life Sciences

Transcription

1 MA 138: Calculus II for the Life Sciences David Murrugarra Department of Mathematics, University of Kentucky. Spring 2016 David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

2 Let s consider N(t) = prey density at time t. P(t) = predator density at time t. David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

3 Let s consider N(t) = prey density at time t. P(t) = predator density at time t. and the following predator-prey model = rn(t) an(t)p(t) = abp(t)n(t) (t) where r, a, b, and d are positive constants. David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

4 Assume that N(t) denotes the density of an insect species at time t and P(t) denotes the density of its predator at time t. The insect species is an agricultural pest, and its predator is use as a biological control agent. Their dynamics are given by the system of differential equations =5N 3PN =2PN P 1 Explain why = 5N describes the dynamics of the insect in the absence of the predator. David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

5 Assume that N(t) denotes the density of an insect species at time t and P(t) denotes the density of its predator at time t. The insect species is an agricultural pest, and its predator is use as a biological control agent. Their dynamics are given by the system of differential equations =5N 3PN =2PN P 1 Explain why = 5N describes the dynamics of the insect in the absence of the predator. Solve = 5N and describe what happens to the insect population in the absence of the predator. David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

6 Assume that N(t) denotes the density of an insect species at time t and P(t) denotes the density of its predator at time t. The insect species is an agricultural pest, and its predator is use as a biological control agent. Their dynamics are given by the system of differential equations =5N 3PN =2PN P 1 Explain why = 5N describes the dynamics of the insect in the absence of the predator. Solve = 5N and describe what happens to the insect population in the absence of the predator. 2 Explain why introducing the insect predator into the system can help to control the density of the insect. David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

7 Solution 1 The solution is N(t) = N(0)e 5t. Thus in the absence of predator, the insect species grows exponentially fast. David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

8 Solution 1 The solution is N(t) = N(0)e 5t. Thus in the absence of predator, the insect species grows exponentially fast. 2 If P(t) > 0, then N(t) stays bounded David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

9 Assume that N(t) denotes prey density at time t and P(t) denotes predator density at time t. Their dynamics are given by the system of differential equations =4N 2PN =PN 3P Assume that initially N(0) = 3 and P(0) = 2. 1 Identify all equilibria of the system. David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

10 Assume that N(t) denotes prey density at time t and P(t) denotes predator density at time t. Their dynamics are given by the system of differential equations =4N 2PN =PN 3P Assume that initially N(0) = 3 and P(0) = 2. 1 Identify all equilibria of the system. 2 If you followed this predator-prey community over time, what would you observe? David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

11 Assume that N(t) denotes prey density at time t and P(t) denotes predator density at time t. Their dynamics are given by the system of differential equations =4N 2PN =PN 3P Assume that initially N(0) = 3 and P(0) = 2. 1 Identify all equilibria of the system. 2 If you followed this predator-prey community over time, what would you observe? 3 Suppose that bad weather kills 90% of the prey population and 67% of the predator population. If you continue to observe this predator-prey community, what would you expect to see? David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

12 Solution 1 The system is in equilibrium at (3, 2). David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

13 Solution The system is in equilibrium at (3, 2) The cycles will increase in magnitude. (3.0,2.0) David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

14 An unrealistic feature of the Lotka-Volterra model is that the prey exhibits unlimited growth in the absence of the predator. To remedy this shortcoming, consider the following model instead, ( 1 N 10 =3N =PN 4P ) 2PN 1 Investigate the long-term behavior in the absence of the predator. David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

15 An unrealistic feature of the Lotka-Volterra model is that the prey exhibits unlimited growth in the absence of the predator. To remedy this shortcoming, consider the following model instead, ( 1 N 10 =3N =PN 4P ) 2PN 1 Investigate the long-term behavior in the absence of the predator. 2 Find all equilibria of the system and use the eigenvalue approach to determine their stability. David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

16 Solution 2.0 (2.0,2.0) 1 The prey evolves according to logistic growth. 2 The equilibrium points are (0, 0) and (4, 9/10) (0.0,0.0) David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

17 Impact of the carrying capacity on the dynamics An unrealistic feature of the Lotka-Volterra model is that the prey exhibits unlimited growth in the absence of the predator. To remedy this shortcoming, consider the following model instead, ( =N 1 N ) 4PN K =PN 5P 1 Draw the zero isoclines of the system for K = 10 and K = 3. David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

18 Impact of the carrying capacity on the dynamics An unrealistic feature of the Lotka-Volterra model is that the prey exhibits unlimited growth in the absence of the predator. To remedy this shortcoming, consider the following model instead, ( =N 1 N ) 4PN K =PN 5P 1 Draw the zero isoclines of the system for K = 10 and K = 3. 2 When K = 10, the zero isoclines intersect, indicating the existence of a nontrivial equilibrium. Analyze the stability of this nontrivial equilibrium. David Murrugarra (University of Kentucky) MA 138: Section Spring / 9

19 Impact of the carrying capacity on the dynamics An unrealistic feature of the Lotka-Volterra model is that the prey exhibits unlimited growth in the absence of the predator. To remedy this shortcoming, consider the following model instead, ( =N 1 N ) 4PN K =PN 5P 1 Draw the zero isoclines of the system for K = 10 and K = 3. 2 When K = 10, the zero isoclines intersect, indicating the existence of a nontrivial equilibrium. Analyze the stability of this nontrivial equilibrium. 3 Is there a minimum carrying capacity required in order to have a nontrivial equilibrium? David Murrugarra (University of Kentucky) MA 138: Section Spring / 9