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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

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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)

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

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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!

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

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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

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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

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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

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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

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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 ?

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

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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

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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

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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

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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

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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

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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.

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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.

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