A SESSION ON CHESSON OR MORE EQUATIONS THAN YOU PROBABLY WANT TO SEE AT 10:30AM Benjamin Adams

Contents  Background  Interest and Role in Community Ecology  His work  The Lottery Model  Variable Environment Theory  The Storage Effect  Scale-Transition Theory  Issues

Background  1974 B.Sc. University of Adelaide, Australia  1978 Ph.D. (Departments of Statistics and Zoology), University of Adelaide, Australia  Postgraduate Research Biologist, University of California, Santa Barbara  Professor of Zoology, Botany and Statistics, Ohio State University  Senior Fellow, Research School of Biological Sciences, Australian National University  Professor, Section of Evolution and Ecology, University of California, Davis  2005-present Professor, Ecology and Evolutionary Biology, University of Arizona

Interest  How organisms interactions and adaptation to variability promotes species diversity and affects ecosystem functioning.

Role  Theoretical Biologist or Mathematical Modeler Chesson and Warner 1981 Chesson and Elner1989

The Work The Lottery Model and derivatives Variable Environmental Theory The Storage EffectScale-Transition Theory

The Work The Lottery Model and derivatives

The Sale’s Lottery System  L is the number of larvae of species i available  c is a constant representing the relative competitive ability of species i Chesson and Warner 1981

Chesson’s Population Model Portion of homes filled by species “i” Surviving individuals in species “i” Dead individuals of all other species Birthrate of “i” divided by the birthrate of the rest Chesson and Warner 1981 Incorporates stochasticity variable among species

Population Model - Non-overlapping i i i i i i i i i ii i i i i i i i ii J J J JJ J J J i i i i J J J J J J J JJ J

i i i i i i i i i ii i i i i i i i ii i i i i i i i i i i i i i iii i i i i i i

Population Model - Overlapping i i i i i i i i i ii i i i i i i i ii J J J JJ J J J i i i i J J J J J J J JJ J

Chesson and Warner 1981

The Work Variable Environmental Theory

Variable Environment Model  r i = Long term growth rate of species i  Δ E = relative mean effect of the environment  Δ C = relative mean effect of competition  Δ I = relative mean of the interactions between environment and competition. Chesson 1989

Δ C – the competition term  Consists of two parts:  The difference between inter- and intraspecific competition independent of fluctuation  The difference between inter- intraspecific competition which is dependent on competition in the previous time period and nonlinear response. Example of nonlinear: two species with different dependence on same resource.

Chesson 1994

ΔI – the interaction term  Composed of three parts  Species specific response to the environment e.g. different responses to temperature fluctuation  Covariance between environment and competition e.g. improved condition increase density thereby increasing demand on resources  The growth rates response to both competition and environmental fluctuation Three different results – additive, subadditive, superadditive  Storage Effect

Chesson 1994

The Work The Storage Effect

 Competing species can coexist when intraspecific competition outweighs interspecific competition  Fluctuation-dependent 3 part mechanism ( ΔI)  Fluctuation must effect birth rate, death rate, or recruitment  Buffered population Examples = seed banks, hibernation, long-lived adults, refuges for spatial storage effect Chesson 1994 Chesson 2000a Chesson 2000b

Buffered population growth  Subadditive results from previous model.

The equation Long term growth rate at low populatons Equalizing mechanism Stabilizing mechanism

Storage effect i i i i i i i i i ii i i i i i i i ii i i i i i i ii i i i iii iii i iii iJJ JJ i i i i i i i i i ii i i i i i i i ii J J J JJ J J J i i i i J J J J J J J JJ J

Test of Storage Effect  Pake and Venable 1995 – Sonoran annuals use seed banks to store maintain populations. Variation due to germination factors.  Caceres 1997 – 30 years of plankton data of two species. Diapaused eggs. Extinction for one definite without storage effect  Sears and Chesson 2007 – use of neighborhood competition to show spatial storage effect.

The Work Scale-Transition Theory

Scale-transition Theory Chesson 2009 Chesson 2011

Scale-transition equation Variability introduced by nonlinearity on population density Variability introduced to local dynamics or physical environment Number of individuals in following time point Chesson 2009

Scale-transition models  Allows you to incorporate emergent variability produced into a model where data is scaled up to a larger perspective.  Produces testable prediction as to how those emergent properties will relate to smaller scale data (i.e. through nonlinearity and variation)  Propose as a potential alternative to meta- community theory for large scale ecological systems

The Work The Lottery Model and derivatives Variable Environmental Theory The Storage EffectScale-Transition Theory

Issues with models  No Allee effects  No extinction possible  No effect of location on dispersal  Deriving actual numbers to represent variables still difficult

References  Caceres, C Temporal variation, dormancy, and coexistence: a field test of the storage effect. Proc. Natl. Acad. Sci. USA 94:9171–75  Chesson P, Warner R Environmental variability promotes coexistence in lottery competitive systems. Am. Nat. 117 (6):  Chesson P, Ellner S Invasibility and stochastic boundedness in monotonic competition models. J. Math. Biol. 27:117–38  Chesson, P A general model of the role of environmental variability in communities of competing species, in “Lectures of Mathematics in Life Sciences,” 20: Amer. Math. Soc., Providence.  Chesson, P Multispecies competition in variable environments. Theo. Pop. Bio. 45:227–276.  Chesson, P. 2000a. Mechanisms of maintenance of species diversity. Ann. Rev. Ecol. System. 31:  Chesson, P. 2000b. General theory of competitive coexistence in spatially varying environments. Theor. Popul. Biol. 58:  Chesson, P Scale transition theory with special reference to species coexistence in variable environments. J. Bio. Dynamics. 3 (2-3):  Chesson, P Scale transition theory: Its aims, motivations and predictions. Ecol. Complex. doi: /j.ecocom  Pake C, Venable L Is coexistence of Sonoran desert annual plants mediated by temporal variability reproductive success. Ecology 76 (1): 246–261.  Sears A, Chesson P New methods for quantifying the spatial storage effect: an illustration with desert annuals. Ecology 88 (9): 2240–2247