Mathematical models combining ecological and genetic approaches in population biology

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Abstract

The review proposes a generalization of ecological and genetic approaches to problems traditionally considered within the framework of mathematical population biology. This approach is not the only possible one, but it seems to us original and promising, since сombining mathematical models of natural selection and population dynamics allows identifying possible mechanisms for the emergence of a complex temporal organization of genetic biodiversity very sensitive to external influences. When taking into account the age structure of populations in models, a multimodality appears, which not only makes it possible to explain the change in the dynamics mode, but also to take a fresh look at general biological ideas about existing patterns in population dynamics. Scenarios for the microevolution of the genetic composition of a population that arise with fluctuating numbers allow to explain and describe the pronounced genetic differentiation of individuals of different generations in populations with a seasonal pattern of reproduction; for example, the origin of differences in genetic structure among successive generations of Pacific pink salmon Oncorhynchus gorbuscha. Such models explain litter size polymorphism well in different (natural and artificial) populations of Arctic foxes Alopex lagopus; as well as the emergence and cessation of fluctuations in the numbers of several rodent species, which have recently been observed in many northern populations of Western Europe (for example, the disappearance of population cycles of voles in a number of populations in Finland and Sweden). The identified features of the dynamic behavior of such systems are important from the point of view of the revision and development of established theoretical concepts, since in such systems the principle of simple combination (superposition) of the results of two models is violated: density-independent natural selection of the best genotypes and density-dependent regulation of population growth; modes appear that were not observed separately in each of the models.

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About the authors

Efim Ya. Frisman

Institute for Comprehensive Analysis of Regional Problems, FEB RAS

Email: frisman@mail.ru
ORCID iD: 0000-0003-1629-2610

Corresponding Member of RAS, Professor, Scientific Director of the Institute

Russian Federation, Birobidzhan

Oksana L. Zhdanova

Institute of Automation and Control Processes, FEB RAS

Author for correspondence.
Email: axanka@iacp.dvo.ru
ORCID iD: 0000-0002-3090-986X

Doctor of Sciences in Physics and Mathematics, Leading Researcher

Russian Federation, Vladivostok

Galina P. Neverova

Institute of Automation and Control Processes, FEB RAS

Email: galina.nev@gmail.com
ORCID iD: 0000-0001-7567-7188

Doctor of Sciences in Physics and Mathematics, Senior Researcher

Russian Federation, Vladivostok

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2. Fig. 1. Dynamics of qn, rn, and xn, obtained from model (3). a – rAA = 3.9;rAa = 2.9;raa = 1.9; f(x) = 1 – X/M; x = X/M; q0 = 0.35; x0 = 0.01. b – rAA = 10;rAa = 30;raa = 2; f(x) = exp(–bX); x = bX; q0 = 0.001; х0 = 1

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3. Fig. 2. Variations in genetic composition in model (5) with linear (a) and exponential (b) functions of limiting the survival of juveniles. ρ = βs/α is the relative contribution of the older age group to limiting the survival process of juveniles. Parameters wAA = sRAA, wAa = sRAa and waa = sRaa in this case, the fertility of the genotypes AA, Aa and aa is characterized taking into account the survival of offspring to adult age class

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4. Fig. 3. Maps of the dynamics regimes: a – genetic composition; b–c – population sizes, supplemented with trajectories (g, d) and phase portraits (e, g) of model (6) for the initial condition x0 = y0 = 1, q0 = 0.99, p0 = 0.01. The numbers correspond to the length of the observed cycle. Index 1 reflects that the population is monomorphic for the a allele, index 2 – the population is monomorphic for the A allele, in areas without an index the genetic composition undergoes biennial oscillations. ρ = α/(sβ) – relative contribution of the younger age group to limiting the reproduction process. Parameters rAA= sRAA, rAa= sRAa and raa in this case characterize the reproductive potentials of the genotypes AA, Aa and aa of the older age class, taking into account the survival of offspring until reaching maturity

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5. Fig. 5. a. Polymorphism existence limits for 3-year cycles with changing m value. Shaded area – polymorphism in case of simple selection; expansion of area due to maternal selection is shaded in gray. b. Polymorphism existence limits for 2–9-year cycles (n is the cycle length) with m = 2/3. Simple selection corresponds to the lower and middle limits of the interval, maternal selection – to the lower and upper limits.

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6. Fig. 6. Predator polymorphism: a – map of dynamic regimes of community population size; b–g – map of predator genetic composition; d–g – examples of dynamics: predator polymorphism and change of dynamic regime in prey population

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