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##### Alan Hastings

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**Alan Hastings**Dept of Environmental Science and Policy, Univ of Calif, Davis two.ucdavis.edu**Lessons from several courses**• Population Biology • Taught from a perspective that emphasizes quantitative approaches and models • Audience is typically college juniors and seniors • Have taken some biology, and at least one quarter of calculus • Today, I’ll look at questions that involve only algebra • Quantitative methods in Population Biology • First year students in graduate school**Population biology**• Quantitative reasoning essential • We count the number of individuals • We count the frequencies of different types in populations • Find that biology students have often not been comfortable with mathematics • Artificial examples • New textbooks may help**Important practical issues**• Epidemics • Hoof and mouth in UK • Childhood diseases • AIDS • Insect pests • Fisheries**Mathematical**step Mathematical step Mathematical step Mathematical step Overall concepts of interaction between math and biology Mathematical formulation Biological conclusion Biological question**Mathematical**step Mathematical step Mathematical step Mathematical step Overall concepts of interaction between math and biology Mathematical formulation Should not be a linear process- feedback Biological conclusion Biological question**Mathematical**step Mathematical step Mathematical step Mathematical step Overall concepts of interaction between math and biology Mathematical formulation Check step with biological reasoning Biological conclusion Biological question**Combining computer and mathematical approaches**• Problems in biology are nonlinear • Involve quadratic and more complex functions • So numerical (computer solutions are essential) • What numerical platform • Easy for students to pick up • Easy for me to put a relatively nice interface on • Spreadsheet, with students just having to enter numbers • Other solutions possible**Then why analytic mathematical reasoning at all?**• Need to think about qualitative behavior • Will population grow (at all) may be much more important question than how fast • Understand and get general results • Get deeper understanding of why • Emphasize interplay among biology, math and numerical/computer results**Geometric or exponential growth**• Question: how do populations grow if resources are unlimited? • (Models are often most useful if their predictions are not upheld) • N(t+1)=R N(t) • What is R? • Exact example – univoltine insects • From this equation we can predict all future population sizes**Geometric growth continued**• N(t+1)=R N(t) • N(t+2)=R N(t+1) • N(t+2)=R (N(t+1))**Geometric growth continued**• N(t+1)=R N(t) • N(t+2)=R N(t+1) • N(t+2)=R (RN(t)) • N(t+2)=R2N(t) • N(t)=RtN(0) • This last formula gives an exact prediction of future population size**Geometric growth continued**N(t)=RtN(0) • This formula gives an exact prediction of future population size • But more interesting: • R> 1 grows • R < 1 declines • R = 1 only way to get equilibrium • Illustrate with Excel**Growth with two age classes**• Build on ideas of previous example • First develop from basic principles • Then use matrices • Stable age structure • Develop analytic solutions • Then numerical ones**Other topics**• Population genetics • Frequency changes in one locus two allele model • Biston betularia example nice • Numerical solutions • Drift • Discuss the example of one individual • Numerical examples • More numerical examples**Other examples**• Epidemics and the threshold theorem • BN/g**Take home messages**• Emphasize tight interplay between biology and quantitative reasoning • Use the simplest analytic models possible • Numerical approaches allow investigation • Develop both mathematical and biological themes – but always focus on the biological question • Emphasize the importance of ‘failures’ of models • two.ucdavis.edu**Anaphes flavipes(Hymenoptera: Mymaridae)**A. flavipes late pupal stage within host. Note the darkened body.PHOTO: USDA, APHIS, PPQ, Niles Plant Protection Center A. flavipes early pupal stage within host. Red compound eyes are the first visible feature.PHOTO: USDA, APHIS, PPQ, Niles Plant Protection Center A. flavipes female on host egg.PHOTO: PHOTO: USDA, APHIS, PPQ, Niles Plant Protection Center**Pleolophus basizonus is an important ectoparasitoid of**diprionids. During outbreaks this species can cause high mortality rates.**Egg of P. basizonus. The eonymph had been paralyzed by the**female parasitoid prior to oviposition.**Pteromalid Wasp Parasitiod of Stable Flyand House Fly**Puparia**Circular hole left by wasp parasitoid emerging from (left)**an armored scale and (right) a soft scale.**Left:Adult female Encarsia inaron.Right: E. inaron exit**holes (arrow) from Ash whitefly nymphs.M.Rose (both)**Braconid Larvae Emergingfrom Mature Red Admiral Caterpillar**– I