Published in: Guex, G.-D., P. Beerli, A. D. Barbour, and H. Hotz. A dynamic model to describe equilibrium conditions in mixed populations of a hemiclonal hybrid and its sexual host in European water frogs. In: Catzeflis, F. M. and M. Gautier (ed.). Evolution 93. Fourth Congress of the European Society for Evolutionary Biology. Montpellier, August 22-28, 1993: 158.


A dynamic model to describe equilibrium conditions in mixed populations of a hemiclonal hybrid and its sexual host in European water frogs

G.-D. Guex (1), P. Beerli (1), A.D. Barbour (2), and H. Hotz (1,3)

1 Zoological Museum, University of Zürich (Switzerland), 2 Institute for Applied Mathematics, University of Zürich (Switzerland), 3 Department of Ecology, Ethology and Evolution, University of Illinios, Urbana (USA)

The long-term prospective of clonal organisms to persist has for two decades been a focus of interest in a central issue of evolutionary biology, the ubiquitous maintenance of sexual recombination despite its lower reproductive efficiency. Hemiclonal systems, in which one parental genome of hybrids is excluded in gametogenesis but replaced in the next generation by fertilization with gametes of the same parental species, suffer the additional constraint of spatial and temporal dependence on this sexual host. In mixed Rana lessonae-Rana esculenta populations, the hybridogenetic hybrid Rana esculenta (RL: R. ridibunda x R. lessonae) uses R. lessonae (LL) as its sexual host. Matings between two hybrids (RL x RL) normally lead to inviable R. ridibunda (RR) larvae. This asymmetry, differences in fecundity parameters, and reported differences in larval performance traits suggest differential reproductive success of RL and LL. To raise further hypotheses on temporal stability of this system, the mathematical space described by life history parameters, and the persistence of the ridibunda genome in R. esculenta lineages far from areas with reproducing R. ridibunda populations, we define a model with a small set of parameters that are measurable in the field: L, E, R = number of reproducing females of R. lessonae, R. esculenta, R. ridibunda resulting from RL x RL matings; a = mean intrinsic fecundity in absence of population pressure; g = intrinsic death rate; k = maximal reproduction possibility of R. lessonae males; K1, K2 = carrying capacity of two different regions in which LL and RL are sheltered from each other except under crowding conditions; population pressures =  and  = , under crowding conditions =  = , and the frequency of RL fertilized by LL, f = . These parameters build a dynamic model:  Under equality conditions we found two biologically interesting different major states of the system, where the fitness measure : (1) sL,1, sE>1: only R. lessonae survive; (2) sL,1, sE,1, >: both regions will fill up with R. lessonae and R. esculenta, respectively. There are two more substates of (2): the first will fill up the region K1 with LL, the region K2 with RL and RR, and in the second the RL and RR will crowd into the region of LL. These two subsystems are not as robust as the one described first. The results give a rather detailed net of conditions that can be found in nature. In peat bogs LL lives with very small fractions of RL, but in other ponds or lakes the frequency of LL drops to about 30%, and in artificial habitats such as gravel pits only few percents of LL occur. The mathematically demanded regions K1 and K2 probably correspond to some microhabitat preferences that separate LL and RL in shared spawning ponds, and to different macrohabitats that provide larger regions differentially suitable for RL and LL. Difference in habitat preferences thus seems to be crucial for the maintenance of a hybridogenetic system. This model provides a background for laboratory and field experiments to investigate the correspondence of ecogenetic to model parameters and to explain the observed long-term stability of R. lessonae-R. esculenta populations in areas were no new clones can be founded by primary hybridization.

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Peter Beerli, Dept. of Genetics, University of Washington, Seattle 98195,