Metapopulation models and Population Outbreaks

Metapopulation is a set of local populations connected by migrating individuals.
Local populations usually inhabit isolated patches of resources, and the degree of isolation may vary depending on the distance among patches F1
Metapopulation models consider local populations as individuals. Dynamics of local populations is either not considered at all, or is considered in a very abbreviated way. Most of metapopulation models are based on colonization-extinction equilibrium.
One of the first metapopulation models was developed by MacArthur and Wilson (1967). They considered immigration of organisms (e.g., birds) from a continent to islands in the ocean. The proportion of islands colonized by a species, p, changes according to the equation F2
Example: There are 100 bird species on the continent and the number of species on islands is in the following table:
Island No. Diameter, km (S) Distance from continent, km (D) Number of bird species Proportion of species (p*) ln(p*/(1-p*))
1 10 30 40 0.40 -0.405
2 3 50 5 0.05 -2.944
3 20 100 20 0.20 -1.386
Using linear regression of ln(p*/(1-p*) against two factors: S and D we get the following model parameters: á = 0.229; â = 0.0467; and ã = -1.29. These parameters can be used to predict the number of species on other islands using information about island size and its distance from the continent.
A similar metapopulation model describes metapopulation dynamics without a continent. In this case, neighboring islands become a source of colonization. The proportion of colonized islands changes according to differential equation F3

Lecture 13. Population Outbreaks
• 1. Ecological mechanisms of outbreaks
• 13.2. A model of an outbreak
• 13.3. Catastrophe theory
• 13.4. Classification of outbreaks
• 13.5. Synchronization of outbreaks in space
Ecological Mechanisms of Outbreaks
Population outbreaks are characterized by rapid change in population density over several orders of magnitude. Only a small number of species have outbreaks: e.g., some insect pests, pathogens, and rodents. Population outbreaks often cause serious ecological and economic problems. Examples of outbreak species: locusts, southern pine beetle, spruce budworm, gypsy moth
Two kinds of outbreaks:
1. Introduction of a species to a new area
2. Growth of a native population.
The second case is most interesting because it is important to understand why a population suddenly increases in its density. Usually an outbreak goes through the following phases F4
Initially ecologists tried to explain outbreaks by direct impact of environmental factors. However, the magnitude of change in these factors was always much smaller than the magnitude of change in population density. Attempts to find a "releasing factor" usually fail.
Thus, there should be an "amlipier" of a small initial disturbance.
Examples of "amplifiers":
1. Inverse density-dependence (positive feedback)
1.1. Escape from natural enemies: gypsy moth.
Mortality caused by generalist predators with a type II or III functional response decreases with increasing prey density. Thus, the greater is the population density, the faster it grows. Escape from natural enemies may also result from a delay in the numerical response of natural enemies (pine sawflies, gypsy moth).
1.2. Group effect: bark beetles, sawyer beetle (Cerambicidae), locusts.
Bark beetles succeed in attacking a healthy tree only when the number of beetles is large. Adults of the sawyer beetle, Monochamus urussovi, feed on small branches of Siberian fir. When the density of adults is high, then they cause considerable damage and the tree looses its resistance to developing larvae of this species. Locusts change their behavior at high population density, and their reproduction rate increases.
2. Density-independent processes.
2.1. Plant response to disturbance: spider mites.
Population of spider mites grow very fast at high temperature. They live on plant leaves where local temperature is lower than the ambient temperature. During the draught, plant transpiration is reduced, and thus, the temperature of leaves increases causing rapid reproduction of spider mites.
2.2. Insect physiological response to disturbance: sawflies.
Pine sawflies, Diprion pini, have >50% of their population in a prolonged diapause lasting from 1 to 5 years. Draught may cause reactivation of a large proportion of diapausing sawflies. This effect is combined with subsequent escape from natural enemies.
These amplifiers can be triggered only at specific state of the population system. When an outbreak is already in progress, additional disturbances have almost no effect. Only when an outbreak cycle is finished, then the population may again respond to another disturbance. In some cases, even small disturbances cause an outbreak, and then the population is permanently in an outbreak cycle.
Outbreaks collapse usually due to one of the following mechanisms:
• Destruction of resources
• Natural enemies
• Unfavorable weather
A model of an Outbreak
The eastern spruce budworm (Choristoneura fumiferana) is a forest pest insect. It defoliates spruce stands in Canada and Maine. Outbreaks occur in intervals of 30-40 years. Last outbreaks were in 1910, 1940, and 1970. They resulted in defoliation of 10, 25, and 55 million hectares, respectively.
Clark and Holling (1979, Fortschr. Zool. 25: 29-52) developed a simple model of the spruce budworm population dynamics. It includes (1) logistic population growth, and (2) type III functional response of polyphagous predators (birds). Biological background of this model is not very solid because several important factors were ignored (e.g., parasitism, diseases). However, the model captures dynamic features of the system and we will use it as an example of an population outbreak models - http://home.comcast.net/~sharov/PopEcol/xls/budworm.xls
Population dynamics is described by the following differential equation F5
Technical details: The second term (predation) is derived from the Holling''s disc equation (lecture 10.3. - Functional and Numerical Response) assuming that the search rate of predators is proportional to prey density (a = qN). Then, alpha = qP, and beta = qTh, where P is the density of predators and Th is handling time.
We will start with the following parameter values:
r = 1; K = 1000; alpha = 0.5; beta = 0.04
This model describes the start of an outbreak due to escape from natural enemies. The phase plot of the model is show below. There are 2 stable equilibria: at the lower equilibrium (N* = 2.5) population numbers are stabilized by predation, and at the higher equilibrium (N* = 2.5) the population reaches its carrying capacity. The switch point between these equilibria is N = 10 , see F6.
This model does not include mechanisms that may cause the collapse of outbreak populations. Thus, the outbreak continues forever in a model population. In nature, outbreak populations of the spruce budworm cause severe defoliation and destroy the forest. As a result, the population collapses. Interaction of spruce budworm with host trees will be considered in the Catastrophe theory
Catastrophe theory was very fashionable in 70-s and 80-s. Rene Thom was one of its spiritual leaders. This theory originated from qualitative solution of differential equations and it has nothing in common with Apocalypse or UFO.
Catastrophe means the loss of stability in a dynamic system. The major method of this theory is sorting dynamic variables into slow and fast. Then stability features of fast variables may change slowly due to dynamics of slow variables.
The theory of catastrophes was applied to the spruce budworm (Choristoneura fumiferana) (Casti 1982, Ecol. Modell., 14: 293-300). We will use the model which was considered in the previous section and modify it by adding a slow variable: the average age of trees in the stand.
The performance of spruce budworm populations is better in mature spruce stands than in young stands. Thus, we will assume that the intrinsic rate of increase (r) and carrying capacity (K) both increase with the age of host trees F7
The age of trees continue increasing with time. Age can be considered as a "slow" variable as compared to population density which is a "fast" variable. Dynamics of the system can be explained using the graph F8
Direction of slow processes. When the density of spruce budworm is low, then there is little mortality of trees and the average age of trees increases. Thus, the slow process at the lower branch of stable budworm density is directed to the right (increasing of stand age). The upper branch of stable budworm density corresponds to outbreak populations. Old trees are more susceptible to defoliation and they die first. Thus, the mean age of defoliated stand decreases, and the slow process at the upper branch goes backwards F9 – http://www.gypsymoth.ento.vt.edu/~sharov/PopEcol/lec13/model.html
We can add stochastic fluctuations due to weather or other factors. As the age of trees increases, the domain of attraction of the endemic (=low-density) equilibrium becomes smaller. As a result, the probability of outbreak increases with increasing forest age. If an outbreak occurs in a young forest stand, then it is possible to suppress the population and return it back to the area of stability. But if the stand is old, then the endemic equilibrium has a very narrow domain of attraction, and thus, the probability of an outbreak is very high. Finally, the lower equilibrium disappears and it is no longer possible to avoid an outbreak by suppressing budworm population.
The model suggests to reduce forest age by cutting oldest trees. This will move the system back into the stable area.
Classification of Outbreaks
• Gradient populations: respond directly to external factors (no density-dependent amplification). They have high density in favorable conditions and low density in unfavorable conditions (both in space and time). High-density populations never spread (cause population increase in surrounding populations).
• Eruptive populations: the effect of external factors is amplified by inverse density-dependence (=release effect). Amplifying mechanisms were discussed in the first section. Outbreaks of eruptive populations are able to spread (traveling wave). See the spreading outbreak of the southern pine beetle.
• Sustained eruption: environmental fluctuations may cause the transition of the population from the low equilibrium to the high equilibrium. Examples: bark beetles, spruce budworm
• Pulse eruption: environmental fluctuations trigger an outbreak which collapses immediately (e.g., due to parasites). Examples: gypsy moth, pine sawflies, etc.
• Cyclical eruption: Both equilibria are unstable and populations cycles around them. Examples: Zeiraphera diniana, Cardiospina albitextura.
In bark beetles and some sawyer beetles Cerambicidae, two stable population equilibria exist because of the positive feedback. Massive beetle attack on a tree overcomes its resistance, and thus, the greater is population density the more resources (weakened trees) are available.
Mechanisms of beetle attack may be different. Bark beetles make holes in the bark. If there are only few beetles, then the holes become filled with resin and beetles die. If thousands of beetles are making their holes simultaneously, then the tree has not enough resin for self-defense.
Sawyer beetles (Monochamus urussovi) oviposit into boles of weakened trees. Adults feed in tree twigs and can weaken a tree if population density is high. As a result, more oviposition sites become available F10

Next
EXEL Outbreak Models
EXEL - Outbreak Models for slow processes F9