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Teaching Biology in Mathematics Classes

Teaching Biology in Mathematics Classes. Glenn Ledder Department of Mathematics University of Nebraska-Lincoln gledder@math.unl.edu funded by NSF grant DUE 0531920 . Overview. Comar, PRIMUS 18, 49-70, 2008 Ledder, PRIMUS 18, 119-138, 2008

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Teaching Biology in Mathematics Classes

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  1. Teaching Biology in Mathematics Classes Glenn Ledder Department of Mathematics University of Nebraska-Lincoln gledder@math.unl.edu funded by NSF grant DUE 0531920

  2. Overview • Comar, PRIMUS 18, 49-70, 2008 • Ledder, PRIMUS 18, 119-138, 2008 • Ledder, Differential Equations: A Modeling Approach, McGraw-Hill, 2005 • Ledder, Mathematical Methods for Biology and Medicine, in preparation

  3. Energy Budget Modeling • What do organisms do with the resources they collect from their food? • Why do different species grow to different sizes? • Do organisms grow to their physiological maximum size?

  4. Energy Budget Modeling Introducing the “kyoob,” a biologically simple creature of cubic shape: • Intake rate is proportional to surface area: 6as2 • Use for tissue maintenance is proportional to volume: bs3 • Surplus resources are used for growth to size S and then reproduction. • Kyoobs live forever. Goals:Find the physiological maximum size and the optimal adult size.

  5. Energy Budget Modeling Surplus Energy: Physiological Maximum: (no surplus) Optimal Size:

  6. Genetics and Evolution • How does natural selection change the gene pool? • Why did natural selection favor the gene for sickle cell anemia? • Should sickle cell anemia disappear in the future? How quickly?

  7. Genetics and Evolution Sickle cell anemia biology: • Everyone has a pair of genes (each either A or a) at the sickle cell locus: • AA: vulnerable to malaria • Aa: protected from malaria • aa: sickle cell anemia • Babies get A from an AA parent and either A or a from an Aa parent.

  8. Let p by the prevalence of A. Let q=1-p be the prevalence of a. Let m be the malaria mortality. Let w(p) be a measure of the relative fitness of the gene pool. w(p)=(1-m) p2 +2p(1-p) 1 1+m Optimum p is ——–.

  9. Things to do with the model: • Explore the action of natural selection when modern medicine changes m to 0. • Find the value q0that yields a sickle cell incidence rate of 4%. [ANSWER: 0.2] • Find the corresponding value m0. [ANSWER: 0.25] • . • Derive a difference equation that determines qt+1 from qt when m=0. [ANSWER: qt+1=qt/(1+qt)] • Solve the difference equation to obtain a sequence for q. [ANSWER: qt=1/(t+5)] • How many generations does it take to reduce sickle cell deaths to 1 in 10,000? [ANSWER: 95]

  10. Demographics / Population Growth • What determines the rate of growth of a population? • How are the ages of members of a population distributed? • In particular, how do changes in birth rates and mortality rates change population growth and structure?

  11. Demographics / Population Growth Let l(x) be the probability of survival to age x. Let m(x) be the rate of production of offspring for parents of age x. Let r be the population growth rate. Let B(t) be the total birth rate. How do l and m determine B and r? • The birth rate should increase exponentially with rate r. • The birth rate can be computed by adding up the births to parents of different ages.

  12. Demographics / Population Growth Population of age x if no deaths: Actual population of age x: Birth rate for parents of age x: Total birth rate at time t: Total birth rate at time t: Euler equation:

  13. Things to do with the model: • Discretize it by assuming that l and m are piecewise constant (integrating over each time interval. • Find real data for l and m. Then use a numerical solver to find r. • Explore the changes in r when • Births are delayed • Parents have fewer babies • Mortality decreases for the elderly • Infant mortality decreases.

  14. Structured Population Dynamics • Can we create a simple model that tracks changes in the age/size/stage distribution of a population? • How do development rates affect population growth?

  15. Presenting Bugbox-population, a 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. larvapupaadult • Boxbugs progress from larva to pupa to adult. • All boxbugs are female. • Larva are born adjacent to their mother.

  16. Structured Population Dynamics The final “bugbox” model: Let Lt be the number of larvae at time t. Let Pt be the number of juveniles at time t. Let At be the number of adults at time t. Lt+1 = sLt+fAt Pt+1 = pLt At+1 = Pt+aAt

  17. Things to do with the model: • Write as xt+1 = Mxt. • Run a simulation to see that x evolves to a fixed ratio independent of initial conditions. • Obtain the problem Mxt= λxt. • Develop eigenvalues and eigenvectors. • Show that the term with largest |λ| dominates and note that the largest eigenvalue is always positive. • Note the significance of the largest eigenvalue. • Use it to predict long-term behavior and discuss its shortcomings.

  18. Computer Simulation Results A plot of Xt/Xt-1 shows that all variables tend to a constant growth rate λ The ratios Lt:At and Pt:Attend to constant values.

  19. Pharmacokinetics • How long does it take before IV medication takes effect? • Why do we have to take some medication once a day and other medication every four hours? • Why does food poisoning last only a short time but lead poisoning lasts forever?

  20. Pharmacokinetics x′ = Q(t) – (k1+r) x + k2y y′ = k1x – k2y Q(t) k1x blood tissues k2y x(t) y(t) rx

  21. Things to do with the model: • Find equilibrium solutions when Q is constant. • Show that the eigenvalues are both negative. • Find realistic parameter values for some medication and run simulations. • Take x(0)=0, y(0)=1, Q(t)=0, r=1, k1=1. What happens for different choices of k2<1? • Relate the result to lead poisoning.

  22. Predator-Prey Dynamics • How do the interactions between predators and prey affect the populations of both? • How does selective killing of predators change the predator and prey populations?

  23. Predator-Prey Dynamics • General idea • x = prey (biomass), y = predator (biomass) • x′ = growth rate without predators • - loss due to predation • y′ =growth rate from predation • - loss rate without prey

  24. Predator-Prey Dynamics • Lotka-Volterra x = prey, y = predator x′ = rx–sxy y′ = esxy – my Predicts oscillations of varying amplitude

  25. Predator-Prey Dynamics • Lotka-Volterra x = prey, y = predator x′ = rx–sxy y′ = esxy – my Predicts oscillations of varying amplitude Predicts impossibility of predator extinction.

  26. Predator-Prey Dynamics • logistic x = prey, y = predator x′ = rx(1 – — )– sxy y′ = esxy – my Predicts stable xy equilibrium if mis small enough x K

  27. Predator-Prey Dynamics • logistic x = prey, y = predator x′ = rx(1 – — )– sxy y′ = esxy – my Predicts stable xy equilibrium if mis small enough and y→0 if m too large x K

  28. Predator-Prey Dynamics • Holling type 2 x = prey, y = predator x′ = rx(1 – — )– —––– y′ = —––– – my x K qxy A + x eqxy A + x

  29. qxy A + x Why —–––? Let s be search rate Let P be predation rate per predator Let f be fraction of time spent searching Let h be the time needed to handle one prey P=fsxand f + hP =1 P =—–––– = —––– sx 1 + shx qx A + x

  30. Predator-Prey Dynamics • Holling type 2 x = prey, y = predator x′ = rx(1 – — )– —––– y′ = —––– – my Predicts stable xy equilibrium if mis small enough. x K qxy A + x eqxy A + x

  31. Predator-Prey Dynamics • Holling type 2 x = prey, y = predator x′ = rx(1 – — )– —––– y′ = —––– – my Predicts stable xy equilibrium if mis small enough and stable limit cycle if m iseven smaller. x K qxy A + x eqxy A + x

  32. Resource Management • Why have natural resources, such as whales or bison, been depleted so quickly? • How can we restore natural resources? • How should we manage natural resources?

  33. Resource Management Let X be the biomass of resources. Let K be the environmental capacity. Let C be the number of consumers. Let G(X) be the consumption per consumer.

  34. Holling type 3 consumption • Saturation and alternative resource

  35. Dimensionless Version k represents the environmental capacity. c represents the number of consumers. Decreasing A increases both k and c.

  36. The resource increases The resource decreases

  37. Stage 1 – natural balance

  38. Stage 2 – depletion Consumption increases to high level.

  39. Stage 3 – inadequate correction Consumption decreases to modest level.

  40. Stage 4 – recovery Consumption decreases to minimal level.

  41. Stage 5 – proper management Consumption increases to modest level.

  42. PRIMUS 18(1), 2008 • Teaching Math to Biology Students: • J.P. Fulton and L. Sabatino, Using the scientific method to motivate biology students to study precalculus • J.D. White and J.P. Carpenter, Integrating mathematics into the introductory biology laboratory course • R.H. Lock and P.F. Lock, Introducing statistical inference to biology students through bootstrapping and randomization • Teaching Biology to Math Students: • T.D. Comar, The integration of biology into calculus courses • R. Burks, J. Lindquist, S. McMurran, What’s my math course got to do with biology? • E. Marland, K.M. Palmer, R.A. Salinas, Biological applications in the mathematics curriculum • L.J. Heyer, A mathematical optimization problem in bioinformatics • Mathematical Modeling: • G. Ledder, An experimental approach to mathematical modeling in biology

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