V. Volpert

On the emergence and evolution of biological species

1809-1882

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

Classical population dynamics

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

1170-1240

Leonhard Euler

1707-1783

An essay on the principle of population

Thomas Malthus

1766-1834

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, whenunchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio.

Preventive growth (Verhulst)

Destructive growth (Lotka-Volterra)

Competition for resources (Darwin)

Reproduction with limited resources (logistic equation)

1804-1849

A. Lotka and V. Volterra

Prey-predator model Competition of species

u – predator

v – prey

Taking into account movement of individuals, we obtain the reaction-diffusion equation

Reaction-diffusion equation

R.A. Fisher, 1890-1962 A.N. Kolmogorov, 1903-1987

I.G. Petrovkii, 1901-1973

N.S. Piskunov

KPP

Fisher – KPP equation (1937-38)

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

Wave propagation (biological invasion)

World population: super exp growth ?

UN estimate

now

Population distribution

Log scale

Logistic growth with space propagation

What happened here?

Recent developments in population dynamics

Local, nonlocal and global consumption of resources

local

nonlocal

global

Nonlocal reaction-diffusion equations

Nonlocal consumption of resources

Morphological space

Intra-specific competition

Local, nonlocal and global consumption of resources

local

nonlocal

global

Darwin’s 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

Question: is it possible to construct biologistically realistic models for whichpopulations behave as in Darwin’s diagram?

Theory of speciation

Stability analysis – Pattern formation

Instability condition: d/( N^2) < const

Britton, Gourley, …

Emergence of structures from a homogeneous in space solution

Periodic wave propagation

Speciation: propagation of periodic waves

Species and families (double nonlocal consumption)

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

Conditions of (simpatric) speciation

Nonlocal consumption of resources (intra-specific competition)

Self-reproduction Diffusion (mutations)

“Phylogenetic” tree of automobiles

Fardier de Cugnot, 1771

(4km/h, 15 min)

Trucks

Passenger cars

Buses

Speciation in science: Mathematics Subject Classification

Partial differential equations

Survival, disappearance and competition of species

Single and multiple pulses

Standing and moving pulses (bistable case)

Moving pulses Evolution with space dependent coefficients

Survival, disappearance and competition of species

Competition of species with nonlocal consumption

Square waves

Survival, disappearance and competition of species

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

Diagram: summary

1 equation

6 equations

Third important case: extinction

Evolution tree of sea shells (ammonites)

External species have more chances to survive

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

How global wealth depends on redistribution

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0 0,02 0,04 0,06 0,08 0,1 0,12

Series1

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0 0,02 0,04 0,06 0,08 0,1 0,12

Series1

Redistribution coefficient

Global wealth increase

Maximal (individual) wealth increase

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

Conclusions

All that we can do, is to keep steadily in mind that eachorganic being is striving to increase at a geometrical ratio;that each at some period of its life, during some season ofthe year, during each generation or at intervals, has tostruggle for life, and to suffer great destruction. When wereflect on this struggle, we may console ourselves with thefull belief, that the war of nature is not incessant, that nofear is felt, that death is generally prompt, and that thevigourous, the healthy, and the happy survive and multiply.

Charles Darwin

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