1. The Final Bugbox Model Let L t be the number of larvae at time t. Let P t be the number of pupae at time t. Let A t be the number of adults at time t. L t +1 = s L t + f A t P t +1 = p L t A t +1 = P t + a A t

2. Discovering Model Behavior We now guide the students through a process of discovery: –The students write Matlab code to run simulations with given parameters and initial data. –We direct the students to the discovery that the proportions always approach a fixed ratio by asking them to plot P / A. We also have them plot X t+1 /X t for each stage.

3. Model Behavior We fix p = a = 0.5, s = 0.25, f = 2, and choose initial population levels. Plot r = P / A. The ratio r approaches a constant that is independent of the initial conditions.

4. Analysis of the Model The stable growth rate is the eigenvalue of largest magnitude. The stable population ratios are given by the corresponding eigenvector. The students find the stable growth rate and population ratios using only basic algebra and “directed ingenuity.”

5. Analysis of the Model “P/A approaches a constant” means that P and A have the same growth rate as t → ∞. Assume L, P, and A all have growth rate λ. L t+1 = λ L t, P t+1 = λ P t, A t+1 = λ A t. λ L t = ¼ L t + 2 A t λ P t = ½ L t λ A t = P t + ½ A t

6. 2 λ ( λ - ¼) P t = 2 A t, P t = ( λ - ½) A t (eqs 1,3) λ P t = ½ L t λ A t = P t + ½ A t Take L t = 2 λ P t (equation 2) Then λ L t = ¼ L t + 2 A t

7. 2 λ ( λ - ¼) P t = 2 A t, P t = ( λ - ½) A t (eqs 1,3) λ P t = ½ L t λ A t = P t + ½ A t Take L t = 2 λ P t (equation 2) Then λ L t = ¼ L t + 2 A t r = ――― ―, r = λ - ½ 1 λ (λ - ¼) (r = P/A)(r = P/A)

8. 2 λ ( λ - ¼) P t = 2 A t, P t = ( λ - ½) A t (eqs 1,3) λ P t = ½ L t λ A t = P t + ½ A t Take L t = 2 λ P t (equation 2) Then λ L t = ¼ L t + 2 A t r = ――― ―, r = λ - ½ 1 λ (λ - ¼) λ (λ - ¼) (λ - ½) = 1 (r = P/A)(r = P/A)

9. r = λ - ½r = λ - ½ A constant ratio is possible only for special values of λ !

10. λ (λ - ¼) (λ - ½) = 1r = λ - ½r = λ - ½ A constant ratio is possible only for special values of λ ! λ ≈ 1.27 r ≈ 0.77 Similarly, we get L/A ≈ 1.96. The growth rate is ≈ 1.27, and the population ratio is ≈ 1.96 : 0.77 : 1.

11. Mathematical Questions Can we improve the calculation method by using a more sophisticated mathematical structure?

12. Mathematical Questions Can we improve the calculation method by using a more sophisticated mathematical structure? What happens if we allow the parameters to take on any biologically- realistic values?

13. Mathematical Questions Can we improve the calculation method by using a more sophisticated mathematical structure? What happens if we allow the parameters to take on any biologically- realistic values? –Are solutions always positive? –Do simulations always tend toward stable ratio and steady growth rate? –Can there be more than one solution? –If so, what do the simulations actually do?

14. Improving the Method Introduce matrix notation. Redo the calculation using matrix notation. –Introduce eigenvalues and eigenvectors. Develop the 2-step solution method: – det ( M - λ I ) = 0 to find the eigenvalues. – ( M - λ I ) x = 0 to find the eigenvectors.

15. Building the Theory We can prove there are no more than n eigenvalues.

16. Building the Theory We can prove there are no more than n eigenvalues. We can prove that long-term behavior is determined by the eigenvalue of largest modulus.

17. Building the Theory We can prove there are no more than n eigenvalues. We can prove that long-term behavior is determined by the eigenvalue of largest modulus. We can state an existence theorem: –Nonnegative, irreducible, primitive matrices have a unique eigenvalue of largest modulus. This eigenvalue is real and its eigenvector is real and strictly positive.

18. Results of Student Work We got excellent lab data to determine model parameters. Population growth data was consistent with model predictions, but with large experimental uncertainty. Predator-prey data was not useful, but we still learned the theory and mathematics.

19. Assessment -- Surveys Big positive changes from pre- to post- –If I need to, I can write a short computer program to study a mathematical model. –A lot of theoretical biology is mathematical. –I can learn a lot of biology from reading articles and books. No negative changes