Population dynamics

The population dynamics of biological populations is the size as well as uniform spatial variation in time. Population dynamics of one or several coupled populations are a prominent subject of biology, especially ecology and theoretical biology. The population dynamics of species both in the longer evolutionary, as well as in ecological time frame determined by multifactorial interactions within the population, with the animate and inanimate environment.

• 3.1 A -species models
• 3.2 multispecies models

• 4.1 population as an isolated system without affecting 4.1.1 Influence of birth
• 4.1.2 Influence of the death rate
• 4.1.3 Influence of birth and death rate

• 4.2.1 independent birth rate and death rate dependent
• 4.2.2 Birth and death rate of population size dependent
• 4.2.3 Mathematical Modeling

• 5.1 predator-prey system
• 5.2 Competitive Model
• 5.3 mutualism

Synecological aspects

Natural, but also the most artificial systems that consist of several species, between which there are different interactions:

• Predator-prey relationships, see Volterra Rules
• Interspecific competition
• Symbioses
• Parasitism

Boundary conditions of the population growth

A distinction is made among the factors that limit the density of a population of two groups:

Density -independent factors

They are independent of the number of individuals that colonize a habitat.

• Weather and weather: The day in - and -week period changing from year to year slightly different climatic events with factors such as temperature, rainfall, wind, sunlight etc.
• Disasters can lead to unpredictable events such as volcanic eruptions, devastating storms, floods, the death of part of the population or the entire population ( local extinction ).
• Non-specific natural enemies: enemies, whose prey spectrum typically includes other living things, and their own population size is thus independent of the random hunted prey.
• Interspecific ( interspecific competition ) competition: The population trends of various animal species in the same habitat with similar demands for food, territories and other resources can be more or less independent from each other, if they occupy different ecological niches.
• Non- infectious diseases: In contrast to infectious diseases that spread more easily in a denser population are non-contagious diseases occurring statistically random events whose relative frequency does not increase with increasing density.
• Pesticides: The use of pesticides ( in agriculture ) results in certain species against which the pesticide is used, the death of part of the population or even the entire population, depending on the amount and intensity of chemical substances used.

Density-dependent factors

The strength of their effect is dependent on the instantaneous population density.

• Intraspecific competition: The competition between individuals in a population to resources such as food, habitat, etc. This depends on the species-specific needs. Thus, individuals need some animal species a large territory, others live in social groups (eg herds or states ) in a smaller space together.
• Social stress ( crowding factor): The coexistence of animals caused by stress encounters and aggression. With increasing density, the stress becomes larger, until at some animal species behavioral changes, infertility or even death occurs.
• Herbivores: animals that serve as prey for predators, so also take influence on their population density. If the number of prey animals, including the predators can raise correspondingly more young, which in turn increase the predation pressure on the prey. (See predator-prey relationship)
• Infectious diseases ( infectious diseases): Everywhere where individuals live in large frames, also increases the risk that an infectious disease spreads rapidly through the transfer of pathogens and thus becomes an epidemic.
• Parasites: At high population density, they can, as well as infectious diseases spread more rapidly.

Basic forms of population dynamics

A species models

A form of density-dependent growth is described by the logistic equation.

Factors are

• The birth rate
• The death rate
• The limiting capacity factor
• The reproductive self-restraint

In nature, this capacity limit is influenced by the following factors:

• Density-independent factors, such as climate

• Density-dependent factors, such as Resources such as food, space,
• Hiding places
• Crowding factor ( social stress)
• Emigration
• Intraspecific competition - intraspecific competition

See also: Gompertz ' refined form of the logistic model and the mathematical derivation of the logistic function.

Growth of an experimental yeast population

• I lag phase (lag engl. to - slow ), start-up phase, zero growth at a low level.

The population size is far from full capacity. The birth rate is approximately equal to the death rate. There is no shortage of resources, density-dependent factors play virtually no role, and there's no competition. The birth rate is low, since at this low population density intraspecific encounters are randomly. The density-dependent mortality from predation plays only a minor role, since there are enough hiding places in the habitat available for the few individuals. Even infectious diseases affect due to the rare intraspecific encounters from very little. Regular fluctuations in population density (oscillations ) are either seasonal or genetic, random fluctuations. However, if a particular population density below the population may die out if no longer held enough meetings reproduction within the breeding season. When a certain value is exceeded, the population can occur in the exponential phase.

• II to IV: phase of positive growth: The birth rate is higher than the mortality rate. The resources are so abundant that intraspecific competition does not matter. The population of predators is initially so small and hiding are so numerous, that the mortality rate remains low. It will take place first exponential growth ( phase II, log phase ), as the birth rate increases faster than the mortality rate.
• With increasing population size but also increases the mortality rate. Rising birth and death rate equal to fast, the population is in the phase III ( linear growth ).
• With increasing proximity to the capacity limit but play intraspecific and interspecific competition possibly an increasingly important role, so that the increase in the birth rate is slowed down and the death rate continues to increase. Further growth is slowed down (phase IV, delayed growth ). With increasing prey density increases, so does the population density of predators, or predators specialize increasingly in this loot.

The high birth rate is offset by a high density-dependent mortality rate ( intraspecific competition, stress, epidemics, etc.). There is maximum occupation of the habitat, the resources are used optimally without exhausting them. The population density varies around the value of K, the so-called environmental capacity. Density -independent factors ( eg seasons) lead to an oscillating or, in unfavorable habitats with fluctuating environmental conditions, fluctuating course. The larger a population is, the more stable, this stationary phase. An overhang increases the death rate and / or lowers the birth rate in the short term. It can be reduced by emigration (example: Lemmings ).

• VI: death phase

The birth rate is now lower than the death rate. In small populations, chance fluctuations can lead to extinction. A reduction of the capacity, for example, by changes in the environment or immigration of new predators can cause the setting of a new equilibrium at a lower level.

Notes:

• In a real population, not all phases occur. For example, accounts for stages II to VI, when the population becomes extinct already from the Phase I out.

• With field observations can take on populations that are already at an advanced stage, so there are already, for example, in Phase IV.

Multispecies models

A relatively simple case arises from interactions between two species that are in a predator-prey relationship. This case is classically described by the Lotka -Volterra rules.

More complex relationships resulting from interaction of several species. There are approaches to describe such relationships using mathematical models and simulated.

Population as a system

The following is to be developed on the basis of systems theoretical considerations mathematical modeling. It is initially assumed that the resources for the population are infinite. Notwithstanding some references birth rate ( birth rate ) and mortality rate are ( mortality rate ) are considered internal system control variables. A population change, for example by birth is therefore not to be taken as a result of the inflow from the outside. ( That is the case rather on immigration ). Interactions with the environment in the form of increase or emigration are also not considered.

• The birth rate is the positive control value, the greater the birth rate is, the larger the population.
• The mortality rate is the negative control value, the greater the death rate is, the smaller the population.

Generally, such systems are described by partial differential equations such as the Kolmogorov equation. In the following special cases of this equation are discussed.

Population as an isolated system without affecting

Influence of birth

It is only the influence of birth viewed on the rate of change of population size N:

Hence ( by integration) under the assumption that the birth rate is constant ( ) for the population size at any time t the output size N (t = 0) = N0:

(1)

This gives a positive linear growth, the growth rate depends only on the birth rate: The higher the birth rate, the faster the growth takes place.

This system can serve as a model for insect colonies (bees, ants, termites ), or other animal populations ( Wolf Pack ), in which only a female gives birth to boy, but neglecting the death rate. This context it should not be forgotten that the modeling results in a continuous growth, but occurs in nature in discrete steps. The birth rate is expected to be constant only temporarily under ideal conditions.

Influence the mortality rate

For the death rate yields:

Const = c2 =. > 0:

(2)

This results in a negative linear growth.

The Extinktionszeit that time, therefore, when the population is extinct, can be calculated with.

In nature there is no equivalent for this purpose, since the death rate always depends on the population size. A regular taking of animals, for example, from a cattle herd would already correspond to the change in an open system.

Influence of birth and death rate

If fertility and mortality rate considered simultaneously, the time variation of the population given by:

Substituting equations (1) and (2) and subsequent integration result:

Although now is still linear growth before, but whether the population is increasing or decreasing or stagnant, depending on the size of the birth and death rate from:

Population as an isolated system with feedback

In nature, birth and death rates depend on the population density and the capacity of the ecosystem.

Independent birth rate and death rate dependent

In the following, the birth rate will continue to be independent of the population size, while the death rate depends on the population density: (The greater the population density, the greater the death rate ). From this proportionality is obtained by means of the proportionality factor c2 = const. > 0 the equation

For the rate of change in population size due to the death rate alone, this resulted in:

(3)

It is obtained by integration:

The inclusion of the constant, independent of the population density birth rate yields:

(4)

The integration results in the equation

This system can go into a different equilibrium state of N0. Balance is achieved when as many as are born to die again, if so, or even if the rate of change in population size is: From equation ( 4), the result for the population size at equilibrium ( equilibrium ):

If the system is in equilibrium from the outset and thus in the steady state. Despite growth through birth and bereavement, the population density does not change, there is zero growth before. Otherwise ( mathematically ), the equilibrium state is reached only with. The date from which you can practically speak of balance, could be calculated using the definition that a minimal difference compared to N ( for example, 0.01 %) is already considered as an equilibrium situation.

Half-life of the equilibrium curve:

Based on the enzyme kinetics could be one time charge equal to the Michaelis constant. Neq t ½ would be the point in time when the population has reached the half of the equilibrium size:

Birth and death rate of population size dependent

Wear in all populations for females at birth, the birth rate of the population density depends on:

(The greater the population density, the greater the birth rate. ) From this proportionality is obtained by means of the proportionality factor c1 = const. > 0 the equation

For the rate of change in population size due to the birth rate becomes:

It is obtained by integration:

Thus results for the rate of change of population size, involving birth and death rate:

It is obtained by integration:

Case distinctions:

This system has no N0 different equilibrium.

In a real population depend, however, fertility and mortality rates not only on the population density, but also on the distance of the population from the capacity limit, which is the the system maximum possible population size, from: The closer a population is at capacity (the smaller so the difference KN), the lower the birth rate and the higher the death rate.

These relationships are described by the simplified model of logistic growth ( Verhulst ):

Mathematical modeling

Basic equation (after Pierre -François Verhulst ):

By integration yields:

Case distinction:

A good agreement of the model with the observations obtained for organisms that reproduce by binary fission (bacteria, yeasts and other eukaryotic unicellular ) and higher organisms, which have a low reproductive rate per generation and the generations overlap.

Are not described fluctuations and oscillations of the population density around the capacity limit as well as decrease the population near the capacity limit.

An extension of the formula of the logistic growth by time delays results in the model also periodic fluctuations:

Models for interacting populations

Predator-prey system

Predator-prey system of Lotka -Volterra equation:

N (t ): Number of prey at time t

P ( t): number of predators at time t,

A, b, c, d > 0: coefficient

Competition model

R (i): linear birth rates

K (i): capacity, limited by resources

B ( 12) B ( 21): competition effects of N2 to N1 or to N2 from N1

Mutualism

Literature: JD Murray, Mathematical Biology BdI, Springer, 2002

Fluctuating population sizes

While the theoretical models contribute to the interpretation of observed fluctuations in fact, they can but these fluctuations do not always unequivocally explain. More or less regular fluctuations are also referred to as oscillations.

Examples of fluctuations:

• The best known example are the lemmings with a turnover rate of 3 to 5 years, but a clear explanation is still pending.
• For the arctic fox in Scandinavia is sometimes a turnover rate of its population size of 3, indicated elsewhere 7-11 and again the other 22-25 years, here the predator-prey system needs to be applied twice: to him as robbers and to him as prey.
• Fluctuations are clearly perceived in European wild animal populations, especially so in feral pigs: In Hochtaunuskreis the years 2000 to 2003 as a very rich wild boar were considered, the years 2004 and 2005 as very wildschweinarm and again in 2008 as extraordinarily rich wild boar. The local press informed extensively about wild boar damage in a few years and lack of wild boar meat in the other. Here perhaps is true, the predator -prey system where the person filling the role of the predator, but more so far unexplained factors also play important roles.