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Chapter 3. Coevolution and Population Dynamics Biological Motivation The previous chapters considered the process of coevolution without touching on changes in population sizes. Although there are many cases where this may be a good approximation of reality, there are clearly other systems where it is unlikely to be true. One obvious example occurs in interactions between infectious diseases and their hosts. In these types of interactions, pathogen populations often go through boom and bust cycles or at least show significant fluctuations in population size over time (REFS). RETURN TO FLAX AND GFG!!! FIRST DO POP GEN THEN DO AD. USE BURDON/THRALL AS MOTIVATION FOR CYCLES. These population fluctuations could have important consequences for coevolution if they cause the intensity or form of reciprocal selection to change over time. COULD ALSO PROVIDE A GENERAL INTRODUCTION POINTING TO EXAMPLES OF PUTATIVE COEVOLVING INTERACTIONS IN WHICH POP SIZES CHANGE. THEN INTRODUCE LV MODEL AND ASK: “WHAT HAPPENS WHEN THE PAREMETERS EVOLVE?COEVOLVE?) WOULD PLANT/PATHOGEN INTERACTION BE SI? this chapter will focus on evaluating the extent to which simple mathematical models of coevolution based on quantitative traits can explain these commonly observed empirical patterns. Key Questions: Does coevolution influence population dynamics? When should coevolution cause population cycles? Can coevolution drive a species to extinction? Building a Model of X-X Coevolution A logical place to begin is by formulating a model that describes the ecology of the interaction between these species. How can we go about defining such a model? The simplest possible approach is to grab a well-studied model from ecology and bend it to our purposes. An obvious candidate is the classical Lotka-Volterra model of predator-prey interactions: = (1 − )− = − (1a) (1b) where is the population abundance of flax, is the population abundance of rust, k is the carrying capacity of the flax population r is the growth rate of the flax population, d is the death rate of rust, is the probability that an encounter between rust and flax leads to death of flax, and is the rate at which Mathematica Resources: http://www.webpages.uidaho.edu/~snuismer/Nuismer_Lab/the_theory_of_coevolution.htm encounters between flax and rust cause infection and production of new rust individuals (REFS). No doubt, you are now wondering why we would use a classical model of a predator prey interaction for the host-pathogen interaction between flax and flax-rust. Analyzing the Model Answers to Key Questions: When should coevolution cause arms races and escalation? What factors determine which species wins? Should we expect coevolution to cause traits of interacting species to match? New Questions Arising: . At the same time, however, our model analyses raise several important questions: What keeps arms races from escalating in perpetuity? Do our results hold for antagonistic interactions with a different mechanistic basis? How would our results differ for other types of ecological interaction? In the next three sections, we will develop generalizations of our simple model which allow us to answer these questions and gain further insight into the process of coevolution mediated by quantitative traits. Generalizations Generalization 1: What keeps arms races from escalating in perpetuity? Generalization 2: Interactions with a different mechanistic basis Generalization #3: Other forms of ecological interaction Conclusions and Synthesis References Figure Legends Figure 1. The interaction function (, ) for three different values of the parameter α. Larger values of α cause the probability of successful oviposition by the weevil to fall off more rapidly as the thickness of the camellia pericarp increases. Figure 2. The exact interaction function (, ) (solid line), and its quadratic approximation (dashed line), for three different values of the parameter α. As the value of α increases, the range of phenotypes over which the quadratic approximation remains accurate decreases. Figure 3. The coevolution of the difference between population mean pericarp thickness in the Camellia and rostrum length in the weevil, = ̅ − ̅ , over 100 generations. The solid line shows a case where > 0, indicating the camellia has a greater response to selection. In this case, pericarp thickness increases more rapidly than rostrum length. The dashed line shows a case where < 0, indicating the weevil has a greater response to selection. In this case, rostrum length increases more rapidly than pericarp thickness. In both cases, coevolution generates sustained arms races where mean rostrum length and pericarp thickness increase in perpetuity. Figure 4. Gaussian stabilizing selection for three different values of the parameter γ. In all cases, the optimal phenotype is θ = 0. Figure 5. The exact fitness function () (solid line), and its quadratic approximation (dashed line), for three different values of the parameter γ. As the value of γ increases, the range of phenotypes over which the quadratic approximation remains accurate decreases. Figure 6. Coevolutionary dynamics of population mean trait values (left hand panels) and corresponding rates of interspecific interaction (right hand panels). In panels a and b, the weevil experiences only very weak selective constraints acting on rostrum length (γX = 0.01), whereas the Camellia pericarp is strongly constrained (γY = 0.06). As a consequence, coevolutionary selection is able to significantly exaggerate rostrum length but not pericarp thickness, thus increasing the average rate of parasitism by the weevil over time. In panels c and d, weevil rostrum length is strongly constrained (γX = 0.06) whereas Camellia pericarp experiences only weak constraints (γY = 0.01). As a consequence, coevolutionary selection is able to significantly exaggerate pericarp thickness but not rostrum length, thus decreasing the average rate of parasitism by the weevil over time. Figure 7. The interaction function (, ) for three different values of the parameter α. Larger values of α cause the probability of successful nest parasitism to fall off more rapidly with differences in cuckoo and warbler egg coloration. Figure 8. The exact interaction function (, ) (solid line), and its quadratic approximation (dashed line), for three different values of the parameter α. As the value of α increases, the range of phenotypes over which the quadratic approximation remains accurate decreases. Figure 9. The coevolution of the difference between population mean egg coloration in warbler and cuckoo, = ̅ − ̅ , over 200 generations. The solid line shows a case where > 0, indicating the warbler has a greater response to selection. In this case, the mean egg coloration of the warbler population increases more rapidly than that of the cuckoo population. The dashed line shows a case where < 0, indicating the cuckoo has a greater response to selection. In this case, the egg coloration of the cuckoo increases more rapidly than that of the warbler population, resulting in a matching equilibrium where the egg coloration of the two species is identical. Figure 10. The interaction function (, ) for three different values of the parameter α. Larger values of α cause the probability of successful consumption and dispersal of nutmeg seeds by the toucan to decline more rapidly as seed size increases relative to beak depth. Figure 11. Coevolutionary dynamics of population mean trait values (left hand panels) and corresponding rates of interspecific interaction (right hand panels). In panels a and b, the nutmeg experiences only very weak selective constraints acting on seed size (γX = 0.01), whereas Toucan beak depth is strongly constrained (γY = 0.06). As a consequence, coevolutionary selection is able to significantly reduce seed size but not beak depth. In panels c and d, nutmeg seed size is strongly constrained (γX = 0.06) whereas Toucan bill depth experiences only weak constraints (γY = 0.01). As a consequence, coevolutionary selection is able to significantly exaggerate bill depth but not significantly reduce seed size. In both cases, the rate at which toucans consume and disperse nutmeg seeds by increases over time because coevolutionary selection drives the mean phenotypes of both species in a direction which promotes the interaction.