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V. Volpert On the emergence and evolution of biological species

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Outline Short history of population dynamics Recent developments: nonlocal consumption of resources Darwin s diagram Theory of speciation Other patterns in the diagram Economical populations

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Classical population dynamics

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First models in population dynamics Population dynamics is one of the oldest areas of mathematical modelling. Already in 1202 Leonard Fibonacci introduced specialsequences of numbers (Fibonacci sequences) in order to describe growth of rabbit population. In 1748 Euler used geometrical sequences (exponential growth) to study human societies. One of the applied problems solved by Leonhard Euler was to verify that the number of people living on Earth at his time could be obtained by a realistic reproduction rate from 6 persons (three sons of Noah and their wives) after the deluge in 2350 BC. Leonard Fibonacci Leonhard Euler

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An essay on the principle of population Thomas Malthus I think I may fairly make two postulata. First, That food is necessary to the existence of man. Secondly, That the passion between the sexes is necessary and will remain nearly in its present state. These two laws, ever since we have had any knowledge of mankind, appear to have been fixed laws of our nature, and, as we have not hitherto seen any alteration in them, we have no right to conclude that they will ever cease to be what they now are.. Assuming then my postulata as granted, I say, that the power of population is indefinitely greater than the power in the earth to produce subsistence for man. Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio. Preventive growth (Verhulst) Destructive growth (Lotka-Volterra) Competition for resources (Darwin)

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Reproduction with limited resources (logistic equation)

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A. Lotka and V. Volterra Prey-predator model Competition of species u – predator v – prey

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Taking into account movement of individuals, we obtain the reaction-diffusion equation Reaction-diffusion equation R.A. Fisher, A.N. Kolmogorov, I.G. Petrovkii, N.S. Piskunov KPP

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Fisher – KPP equation ( ) F(u)=u(1-u) Existence for all speeds > or = minimal velocity Global convergence to waves u(x,t) = w(x-ct) w + c w + F(w) = 0

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Wave propagation (biological invasion)

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World population: super exp growth ? UN estimate now Population distribution Log scale Logistic growth with space propagation What happened here?

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Recent developments in population dynamics

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Local, nonlocal and global consumption of resources local nonlocal global

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Nonlocal reaction-diffusion equations

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Nonlocal consumption of resources Morphological space Intra-specific competition

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Local, nonlocal and global consumption of resources local nonlocal global

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Darwins diagram and its mathematical interpretation Let A to L represent the species of a genus large in its own country; these species are supposed to resemble each other in unequal degrees, as is so generally the case in nature, and is represented in the diagram by the letters standing at unequal distance... The little fan of diverging dotted lines of unequal length proceeding from (A), may represent its varying offspring. phenotype population density

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Question: is it possible to construct biologistically realistic models for which populations behave as in Darwins diagram?

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Theory of speciation

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Stability analysis – Pattern formation Instability condition: d/( N^2) < const Britton, Gourley, …

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Emergence of structures from a homogeneous in space solution

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Periodic wave propagation

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Speciation: propagation of periodic waves

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Species and families (double nonlocal consumption)

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Some remarks 1. Existence, stability, structure of waves, nonlinear dynamics 2. Total mass of the periodic structure is greater than for the constant solution emergence of new species allows more efficient consumption of resources

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Conditions of (simpatric) speciation Nonlocal consumption of resources (intra-specific competition) Self-reproduction Diffusion (mutations)

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Phylogenetic tree of automobiles Fardier de Cugnot, 1771 (4km/h, 15 min) Trucks Passenger cars Buses

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Speciation in science: Mathematics Subject Classification Partial differential equations

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Survival, disappearance and competition of species

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Single and multiple pulses

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Standing and moving pulses (bistable case) Moving pulses Evolution with space dependent coefficients

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Survival, disappearance and competition of species

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Competition of species with nonlocal consumption

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Square waves

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Survival, disappearance and competition of species

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Cold war model Species u moves to decrease its mortality; it consumes resources of species v when their phenotypes are close; species v tries to escape; it increases its global consumption and disappears

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Diagram: summary 1 equation 6 equations

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Third important case: extinction Evolution tree of sea shells (ammonites) External species have more chances to survive

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Economical populations u(x,t) – distribution of wealth Production of wealth is proportional to the value of wealth and to available resources Diffusion – redistribution of wealth Large d: homogeneous wealth distribution Small d: nonhomogeneous wealth distribution

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How global wealth depends on redistribution Redistribution coefficient Global wealth increase Maximal (individual) wealth increase

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Increasing redistribution we get homogeneous wealth distribution (no rich and poor) But the total wealth of the society is greater in the case of nonhomogeneous distribution is greater (capitalism is economically more efficient) Malthus: The powerful tendency of the poor laws to defeat their own purpose Economical populations: some conclusions

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Conclusions All that we can do, is to keep steadily in mind that each organic being is striving to increase at a geometrical ratio; that each at some period of its life, during some season of the year, during each generation or at intervals, has to struggle for life, and to suffer great destruction. When we reflect on this struggle, we may console ourselves with the full belief, that the war of nature is not incessant, that no fear is felt, that death is generally prompt, and that the vigourous, the healthy, and the happy survive and multiply. Charles Darwin

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Acknowledgments and references Properties of integro-differential operators, existence of waves – N. Apreutesei, I. Demin, A. Ducrot Spectrum, stability of waves – A. Ducrot, M. Marion, V. Vougalter Numerical simulations - N. Bessonov, N. Reinberg Biological applications – S. Genieys

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