Predator-prey models Matthew Macauley Department of Mathematical - PowerPoint PPT Presentation

Predator-prey models Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4500, Spring 2016 M. Macauley (Clemson) Predator-prey models Math 4500, Spring 2016 1 / 11

Introduction Consider a population of two species, e.g., foxes (“predator”) and rabbits (“prey”). P t ✏ size of prey. Q t ✏ size of predator. The change in population size of each is a function of both population sizes: ∆ P ✏ F ♣ P , Q q , ∆ Q ✏ G ♣ P , Q q . Question What would happen if the predator or the prey disappeared? Prey, without predators: ∆ P ✏ r ♣ P ♣ 1 ✁ P t ④ M qq . Predators, without prey: ∆ Q ✏ ✁ uQ , where u P ♣ 0 , 1 q is per-capita death rate. Simple predator-prey model ★ ∆ P ✏ rP ♣ 1 ✁ P ④ M q ✁ sPQ r , s , u , v , K → 0, u ➔ 1 ∆ Q ✏ ✁ uQ � vPQ M. Macauley (Clemson) Predator-prey models Math 4500, Spring 2016 2 / 11

Predator-prey model Alternate form ★ P t � 1 ✏ P t ♣ 1 � r ♣ 1 ✁ P ④ M qq ✁ sP t Q t r , s , u , v , K → 0 , u ➔ 1 Q t � 1 ✏ ♣ 1 ✁ u q Q t � vP t Q t The ✁ sPQ and vPQ are called mass-action terms . Roughly speaking: ✁ sPQ describes a negative effect of the predator-prey interaction on the prey, vPQ describes a positive effect of the predator-prey interaction on the predator. Qualitatively, larger values of s and v indicate stronger predator-prey interaction. We can plot the solutions of these equations several ways: time plots: P t vs. t , and Q t vs. t phase plots: Q t vs. P t . M. Macauley (Clemson) Predator-prey models Math 4500, Spring 2016 3 / 11

Time plots and phase plots Consider the following predator-prey model: ★ P t � 1 ✏ P t ♣ 1 � 1 . 3 ♣ 1 ✁ P t qq ✁ . 5 P t Q t Q t � 1 ✏ . 3 Q t � 1 . 6 P t Q t Solutions can be graphed using a time plot (left) or a phase plot (right): M. Macauley (Clemson) Predator-prey models Math 4500, Spring 2016 4 / 11

Equilibria To find steady-state population(s), we set P t ✏ P t � 1 ✏ P ✝ and Q t ✏ Q t � 1 ✏ Q ✝ . P ✝ ✏ P ✝ ♣ 1 � 1 . 3 ♣ 1 ✁ P ✝ qq ✁ . 5 P ✝ Q ✝ ★ ★ P t � 1 ✏ P t ♣ 1 � 1 . 3 ♣ 1 ✁ P t qq ✁ . 5 P t Q t ù Q ✝ ✏ . 3 Q ✝ � 1 . 6 P ✝ Q ✝ Q t � 1 ✏ . 3 Q t � 1 . 6 P t Q t Via simple algebra, this reduces to the following system 0 ✏ P ✝ ♣ 1 . 3 ✁ 1 . 3 P ✝ ✁ . 5 Q ✝ q ★ 0 ✏ Q ✝ ♣✁ . 7 � 1 . 6 P ✝ q If Q ✝ ✏ 0, then P ✝ ✏ 0 or P ✝ ✏ 1. Alternatively, P ✝ ✏ . 4375, which would force Q ✝ ✏ 1 . 4625. Thus, there are three equilibria: ♣ P ✝ , Q ✝ q ✏ ♣ 0 , 0 q , ♣ 1 , 0 q , ♣ . 4375 , 1 . 4625 q . M. Macauley (Clemson) Predator-prey models Math 4500, Spring 2016 5 / 11

Equilibria and nullclines For the general predator-prey model: ★ P t � 1 ✏ P t ♣ 1 � r ♣ 1 ✁ P t ④ M qq ✁ sP t Q t r , s , u , v , K → 0 , u ➔ 1 Q t � 1 ✏ ♣ 1 ✁ u q Q t � vP t Q t the equilibrium equations (set P t ✏ P t � 1 ✏ P ✝ and Q t ✏ Q t � 1 ✏ Q ✝ ) are ★ 0 ✏ P ✝ ♣ r ♣ 1 ✁ P ✝ q ✁ sQ ✝ q 0 ✏ Q ✝ ♣✁ u � vP ✝ q . For Equation 2 to be satisfied, Q ✝ ✏ 0 or ✁ u � vP ✝ ✏ 0. Furthermore, Equation 1 is satisfied if P ✝ ✏ 0 or r ♣ 1 ✁ P ✝ q ✁ sQ ✝ ✏ 0. By simple algebra, we get three equilibria: ✁ u v , r 1 ✁ u ✟✠ ♣ P ✝ , Q ✝ q ✏ ♣ 0 , 0 q , ♣ 1 , 0 q , � . s v A nullcline is a line on which either ∆ P ✏ 0 or ∆ Q ✏ 0. In our example: Q ✏ r P ✏ u P ✏ 0 , s ♣ 1 ✁ P q , Q ✏ 0 , v . M. Macauley (Clemson) Predator-prey models Math 4500, Spring 2016 6 / 11

Nullclines We can plot the nullclines on the PQ -plane to help visualize the dynamics. Q P ✏ u v r Q ✏ r s ♣ 1 ✁ P q s ∆ P ➔ 0 ∆ P ➔ 0 ∆ Q ➔ 0 ∆ Q → 0 ✌ ∆ P → 0 ∆ Q ➔ 0 ∆ P → 0 ∆ Q → 0 P ✌ ✌ 1 ∆ Q ✏ 0 ∆ P ✏ 0 ∆ P → 0 occurs below Q ✏ r s ♣ 1 ✁ P q . ∆ Q → 0 occurs to the right of P ✏ u v . Do you see how we determine the direction of the green arrows? Can we tell whether it spirals inward or outward? M. Macauley (Clemson) Predator-prey models Math 4500, Spring 2016 7 / 11

Nullclines Remark Changing r or s doesn’t affect the Q -nullcline. Q P ✏ u r Q ✏ r v s ♣ 1 ✁ P q s ✌ ♣ P ✝ , Q ✝ q P ✌ ✌ 1 ∆ P ✏ 0 ∆ Q ✏ 0 Suppose the predator was an insect and the prey was an agricultural crop. One might want to introduce a new crop variety with higher r , to try to “outgrow” the predator. Unfortunately, this won’t work: P ✝ is unchanged, but Q ✝ increases. (Why?) M. Macauley (Clemson) Predator-prey models Math 4500, Spring 2016 8 / 11

Linearization Suppose ♣ P ✝ , Q ✝ q is a fixed point whose stability we wish to understand. We can plug the following “perturbation” back into the original system: P t ✏ P ✝ � p t , P t � 1 ✏ P ✝ � p t � 1 , Q t ✏ Q ✝ � q t , Q t � 1 ✏ Q ✝ � q t � 1 . Consider the fixed point ♣ P ✝ , Q ✝ q ✏ ♣ . 4375 , 1 . 4625 q of our previous example. Plugging P t ✏ . 4375 � p t , P t � 1 ✏ . 4375 � p t � 1 , Q t ✏ 1 . 4625 � q t , Q t � 1 ✏ 1 . 4625 � q t � 1 . ★ P t � 1 ✏ P t ♣ 1 � 1 . 3 ♣ 1 ✁ P t qq ✁ . 5 P t Q t into Q t � 1 ✏ . 3 Q t � 1 . 6 P t Q t and simplifying yields ★ p t � 1 ✏ . 43125 p t ✁ . 21875 q t ✁ 1 . 3 p 2 t ✁ . 5 p t q t q t � 1 ✏ 2 . 34 p t � q t � 1 . 6 p t q t For small perturbations ♣ p t , q t q , we can neglect the nonlinear terms (e.g., p 2 t , q 2 t , and p t q t ) which are ✓ 0, leaving a linear system p t � 1 ✓ Ap t . M. Macauley (Clemson) Predator-prey models Math 4500, Spring 2016 9 / 11

Linearization (cont.) Thus, given a small perturbation ♣ p t , q t q at time t , it can be described at time t � 1 by a linear equation p t � 1 ✓ Ap t : ✒ ✚ ✒ . 43125 ✁ . 21875 ✚ ✒ ✚ p t � 1 p t ✓ . 2 . 34 1 q t � 1 q t The eigenvalues of A are λ ✏ . 7156 ✟ . 6565 i , which have norm ♣ . 7156 q 2 � ♣ . 6565 q 2 ✏ . 9711 ➔ 1 . ❛ ⑤ λ ⑤ ✏ Thus, this perturbation from the steady-state is shrinking. The population will spiral back into the steady-state ♣ P ✝ , Q ✝ q ✏ ♣ . 4375 , 1 . 4625 q . Types of equilibrium points ⑤ λ 1 ⑤ ➔ 1, ⑤ λ 2 ⑤ ➔ 1, stable ⑤ λ 1 ⑤ → 1, ⑤ λ 2 ⑤ → 1, unstable ⑤ λ 1 ⑤ ➔ 1, ⑤ λ 2 ⑤ → 1, saddle M. Macauley (Clemson) Predator-prey models Math 4500, Spring 2016 10 / 11

Other interaction models Competition: 2 species fill the same niche in an environment. ★ ∆ P ✏ rP ♣ 1 ✁ ♣ P � Q q④ K q ∆ Q ✏ rQ ♣ 1 ✁ ♣ P � Q q④ K q Question : Does one species “win”? Or can the co-exist? ★ ∆ P ✏ rP ♣ 1 ✁ ♣ P � Q q④ K q ✁ sPQ Competition with predator/prey: ∆ Q ✏ rQ ♣ 1 ✁ ♣ P � Q q④ K q ✟ vPQ ★ ∆ P ✏ rP ♣ 1 ✁ P ④ K q � sPQ Mutualism: e.g., P ✏ sharks, Q ✏ feeder fish. ∆ Q ✏ ✁ uQ � vPQ Immune system vs. infective agent: ★ P : immune cells ∆ P ✏ rQ ✁ sPQ Q : level of infection ∆ Q ✏ uQ ✁ vPQ ✁ sPQ : negative effect on immune system from fighting ✁ sPQ : limited effect on immune system from fighting rQ : immune response is proportional to infection level M. Macauley (Clemson) Predator-prey models Math 4500, Spring 2016 11 / 11