1 The topic 2 Modeling 2.1 Numerical models Evolution of a population

Population can be defined as a set of individual organisms of one species, inhabiting a certain area and a certain space. Species is a class of organisms or individuals which has some commom characteristics or qualities. Species is interbreeding, which means they can breed together and produce fertile descendants. Individual organisms can be divided into two types: Unitary organisms, those proceeded in determinate way. E.g. mammals, birds, insects… The contrast is modular organisms which grow in indeterminate path. E.g. herbs, fungi, coral…

Figure2.1.1

Influences on population size Population size is the number of individual organisms in a population. As it shown in Figure2.1.2, population size is influenced by birth, death and movement of the population. Individual organism can emigrate and immigrate from a certain area.

Figure 2.1.2

Life history (of unitary organisms) Unitary organisms grow in sequence phase. The basic patterns include juvenile phase which indicate the growth of individual organism, reproductive phase and post-reproductive phase. In juvenile period, individual organisms grow into adults and become productive. They produce their descendants in large scale in reproductive phase. After this period, they go into post-reproductive phase and die. Figure2.1.3 illustrates the reproductive output of unitary organisms.

Figure 2.1.3

Motion of organisms Dispersal is the motion of individuals of a species relative to each other. Species move away from the existing population or their parent organism. Such as the seeds of trees, male elephants with respect to the herd. Dispersal also affects spatial distribution like flying taraxacum. Dispersal of organisms can be density-dependent. Individuals may move away from high density because the declining resourses, or they move away from low density to avoid inbreeding. Migration, on the other hand, is directed mass movement of individuals. For example, if original place is lack of water, elephant herd will move to a place with water resources; Eels try to return Sargasso Sea to lay eggs. Migration has certain impact on population and individuals migrate mainly from the declining resourse.

Qualitative types of life histories Plants and animal species reproduce at least once in their lifetime. Some species can repeated breeding (reproduce) multiple times as they age such as humans, carabid beatles, and oak trees. Species that reproduce multiple times are called iteroparous species. Iteroparous species can be again devided into types: seasonal iteroparous which will reproduce by season cycles, such as mouse, birds. Continuous iteroparous that are not depend of season, such as human. Other species, on the other hand, only reproduce once are called semelparouos species. In general, iteroparous species are more common than semelparous species.

Figure 2.1.4

Population growth with intraspecific competition  Wikipedia defines population growth as the change in population over time, and can be quantified as the change in the number of individuals in a population using “per unit time” for measurement. Set N to be the number of individuals in population, N(t) defines the population function over time, dN/dt is the population change over time. Here, new individuals is not included the individual by transformation of material, instead, it only indicate the reproduction of existing individuals. Now we are going to draw the function of the population growth, r represents intrinsic rate of natural increase, such as reproduction rate per individual, change of the population over time dN/dt equals to r multiply the individual number N, and we can get the function N(t) = er*t , see Figure 2.1.5.

Figure 2.1.5 Figure 2.1.5 point out that population will grow ultimately, but is not like this in reality. Let’s see the observed population of trout in Figure 2.1.5.

Figure 2.1.5

Population growth with intraspecific competition Since the resources are limited and the population is growing, the members inside the species will compete with each other over these certain resources, such as food, water, light, space, safe sites, or mates. This is so called intraspecific competition. Members maybe die of starvation or move out to a new place with enough resources. And it will also influence the production of offspring by female. So we can say that net rate equals r only for small population. When the population exceeds some certain boundary, the population size will stop increasing or even decreasing. Look at figure 2.1.6,we assume the net rate r is linear decreasing. K is the maximal capacity the specie can grows, we have the function: 1/N*dN/dt = r - (r/K)N. And according to this function, we get the “logistic equation”: dN/dt = rN*[1 – (N/K)].

Figure 2.1.7

Intraspecific Competition Intraspecies competition can be indirectly, via resource depletion, for example, the grass eaten by an elephant cannot be eaten by others, and we call this exploitation competition. And on the other hand, competition can be directly, actively, by fighting, it is called interference competition. The levelling of population growth at high densities (known as density dependent inhibition) can be seen as an effect of intraspecific competition. Indeed, whereas at low densities organisms do not compete for resources because everybody is satisfied and has enough resource,they do not need to fight, at higher densities resources become limiting, and the population size can no longer increase.

“Capturing the essence of ecological processes” “ … a pattern generated by such a model … is not of interest, or important, because it is generated by the model. … Rather, the point about the pattern is that it reflects important, underlying ecological processes – and the model is useful in that it appears to capture the essence of those processes.” (Townsend et al., Essentials of Ecology) Model like this dN/dt = rN*[1 – (N/K)] may reflect the processes, but exact meaning is still implicit only based on this model, like in the figure 2.1.7, we cannot figure out where is “birth”, “competition”,”death”…