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

Overall concepts of interaction between math and biology Biological question Mathematical formulation Mathematical step Mathematical step Mathematical step Mathematical step Biological conclusion

Overall concepts of interaction between math and biology Biological question Mathematical formulation Mathematical step Mathematical step Mathematical step Mathematical step Biological conclusion Should not be a linear process- feedback

Overall concepts of interaction between math and biology Biological question Mathematical formulation Mathematical step Mathematical step Mathematical step Mathematical step Biological conclusion Check step with biological reasoning

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)=R 2 N(t) N(t)=R t N(0) This last formula gives an exact prediction of future population size

Geometric growth continued N(t)=R t N(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

Mud turtle

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 female on host egg. PHOTO: 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 late pupal stage within host. Note the darkened body. 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 Fly and House Fly Puparia

Circular hole left by wasp parasitoid emerging from (left) an armored scale and (right) a soft scale.

At left, larval and (right), pupal stages of a parasitoid.

Left: Adult female Encarsia inaron. Right: E. inaron exit holes (arrow) from Ash whitefly nymphs. M.Rose (both)

Braconid Larvae Emerging from Mature Red Admiral Caterpillar – I