Multispecies Models (Introduction to predator-prey dynamics)

1. Multispecies Models(Introduction to predator-prey dynamics) Fish 458, Lecture 26

2. Overview • All of the models examined so far ignore multispecies considerations. • We can divide multispecies considerations into biological and technological interactions.

3. Biological and Technological Interactions • Technological Interactions: linkage among species occurs because of their co-occurrence in catches. • Biological Interactions: linkage among species occurs because one eats the other or they compete for the same prey. • This lecture and the next lecture will focus on biological interactions.

4. Biological Interactions • We will develop our models of biological interactions using lumped differential equations (i.e. we are modelling the rate of change of population size / biomass). • Multi-species / eco-system models are, however, extremely complicated and we will quickly have to resort to numerical methods to make use of them.

5. Example 1 : Foxes and Rabbits • In the absence of foxes, rabbits increase uncontrolled while in the absence of rabbits, foxes die due to starvation: • Now let the foxes prey on the rabbits and see what happens:

6. Example 1 : Foxes and Rabbits

7. How Did We Do That? • Simple method: • Keep h very small. However, this simple approach can be very inaccurate. • I used the Runge-Kutta method – it is much more accurate (and pretty fast).

8. Understanding Predator-prey Dynamics • The properties of the predator-prey system can be worked out from the form of the differential equation – the phase diagram. • The population trajectories will often be strongly impacted by the initial conditions.

9. Constructing a Phase Diagram • Find any equilibrium points, i.e, values of y such that: • Draw the isoclines – lines for which the derivative is zero for one of the variables. • Draw arrows on each isocline indicating the rate of change of all other variables.

10. Back to Foxes and Rabbits • The equilibrium point is (F=1,R=1). • The isoclines are defined by: Isoclines

11. Adding in Rates of Change Rabbits unchanging Foxes increasing

12. But Rabbit Populations don’t Grow Forever! • We will extend the model by allowing for some density-dependence in the growth rate for the rabbit population, i.e.:

13. This stablizes the population

14. Computing the Phase Diagram • We proceed as before • Compute the equilibrium point (R=1;F=0.8). • Compute the iscolines:

15. The Phase Diagram-I The point (R=1;F=0.8) is a stable equilibrium t=0

16. The Phase Diagram-II

17. Feeding Functional Relationships - I • The current model assumes that the amount consumed per capita is related linearly to the amount of the prey. • This may be realistic at low prey population size but there must be predator saturation. • We model this effect using feeding functional relationships

18. Feeding Functional Relationships - II

19. Feeding Functional Relationships - III

20. Multispecies Models • Advantages: • Predator-prey dynamics are clearly realistic! • Managers are often interested in “ecosystem considerations”.

21. Multispecies Models • Disadvantages: • It is very difficult to select functional forms / the number of species. • The number of parameters in a multispecies model can be enormous (‘000s). • The results of multispecies models are often sensitive to their specifications. • The methods required to conduct the numerical integrations can be complicated and, if not done correctly, numerical integration impacts the results markedly.

22. Readings • Press et al. (1988), Chapter 16. • Starfield and Bleloch, Chapter 6.