1. Population Growth in a Structured Population Glenn Ledder University of Nebraska-Lincoln http://www.math.unl.edu/~gledder1 gledder@math.unl.edu Supported by NSF grant DUE 0536508

2. Population Growth Unstructured population model: a model that counts all individuals together (discrete exponential function b t ) Structured population model: a model that counts individuals by category (not an elementary mathematical function)

3. Outline 1.Introduce mathematical modeling. 2.Introduce the mathematical model concept. 3.Use unstructured population growth as an example. 4.Model structured population growth.

4. Mathematical Modeling Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation A mathematical model represents a simplified view of the real world. We want answers for the real world. But there is no guarantee that a model will give the right answers!

5. Mathematical Model Input DataOutput Data Key Question: What is the relationship between input and output data? Mathematical Model

6. Unstructured Population Growth -- Approximation Tomorrow’s population depends only on today’s population. All individuals alive tomorrow are born today or survive from today to tomorrow. Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation

7. Unstructured Population Growth -- Derivation Fecundity & Survival Growth Rate & Population Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation N t & N t+1 : today’s and tomorrow’s populations f & s : fecundity and survival parameters

8. Unstructured Population Growth -- Analysis Fecundity & Survival Growth Rate & Population N t+1 /N t = f + s Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation N t = N 0 (f + s) t

9. Unstructured Population Growth -- Validation Misses elements of chance. Misses environmental limitations. Pretty good for a short-time average. Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation

10. Structured Population Growth Some populations have distinct reproductive and non-reproductive stages. 1.Can we make a model for a structured population? 2.Will we find N t+1 /N t = f + s ?

11. Getting Started A conceptual model requires scientific insight. We should observe experiments. Experiments for structured population growth are tricky, expensive, and time- consuming.

12. Presenting Bugbox-population, a real biology lab for a virtual world. http://www.math.unl.edu/~gledder1/BUGBOX/ Boxbugs are simpler than real insects: – They don’t move. – Development rate is chosen by the experimenter. – Each life stage has a distinctive appearance. Boxbugs progress from larva to pupa to adult. All boxbugs are female. Larva are born adjacent to their mother. larva pupa adult

13. Structured Population Dynamics Species 1: Let L t be the number of larvae at time t. Let P t be the number of juveniles at time t. Let A t be the number of adults at time t. L t+1 = f A t P t+1 = L t A t+1 = P t

14. Structured Population Dynamics Species 2: Let L t be the number of larvae at time t. Let P t be the number of juveniles at time t. Let A t be the number of adults at time t. L t+1 = f A t P t+1 = p L t A t+1 = P t

15. Structured Population Dynamics Species 3: Let L t be the number of larvae at time t. Let P t be the number of juveniles at time t. Let A t be the number of adults at time t. L t+1 = f A t P t+1 = p L t A t+1 = P t + a A t

16. Structured Population Dynamics Species 4: Let L t be the number of larvae at time t. Let P t be the number of juveniles 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

17. Computer Simulation Results A plot of X t / X t-1 shows that all variables tend to a constant growth rate λ The ratios L t :A t and P t :A t tend to constant values.

18. Equation for Growth Rate k (k-a) (k-s) = pf N t+1 / N t → k (constant) There is always a unique k that is larger than both a and s.