Lecture 12. Dispersal and Spatial Dynamics

Lecture 12. Dispersal and Spatial Dynamics

F2 - Diffusion model - http://home.comcast.net/~sharov/PopEcol/xls/diffus.xls

F3 Random walk can be defined in a 2-dimensional space. If organisms were released at the center of coordinates (0,0), then their distribution can be described by 2-dimensional normal distribution

This is a 2-dimensional normal distribution:

where N is population density, and D is diffusion coefficient. This equation indicates that the rate of population change is proportional to the curvature of population density. Examples below show that population increases where curvature is positive and decreases where it is negative

Skellam (1951) combined diffusion equation in a 2-dimensional space with exponential local population growth:

F6 Model of Skellam

Skellam''s model gives the following equation for the rate of population expansion

F7 -where M(t) is mean displacement of organisms recaptured t units time after their release (Skellam 1973).

Stratified Dispersal

Gyp Moth

F8 - Population numbers in a colony, N(a), increases exponentially with colony age, a

where no are initial numbers of individuals in a colony that has just established, and r is the marginal rate of population increase.

F9, where n(a) is the number of individuals in a colony of age a. The population front is defined by the condition N(0) = K. Thus, the traveling wave equation is

This equation can be used for estimating the rate of population spread. To estimate this integral we need to define explicitly functions b(x) and n(a). We assume a linear function of the rate of colony establishment:

F10, where V = v/xmax is the relative rate of population spread. This equation can be solved numerically for V; and then the rate of spread is estinmated as v = V xmax.

F11

F12 - This model predicted that barrier zones used in the Slow-the-Spread project should reduce the rate of gypsy moth spread by 54%. This prediction was close to the 59% reduction in the rate of gypsy moth spread in Central Appalachian Mountains observed since 1990 (when the strategy of eradicating isolated colonied has been started).

F13 - Cellular automata model - the figure shows 3 basic rules for the dynamics of cellular automata that simulates stratified dispersal: Stochastic long-distance jumps, Continous local dispersal, Population growth (population numbers are multiplied by R)

Results of cellular automata simulation for several sequential time steps. It is seen how isolated colonies become established, grow, and then coalesce. This model was used for prediction of barrier-zone effect on the rate of population spread and the results were similar to those obtained with the metapopulation model..

12.1. Random Walk
12.2. Diffusion Models
12.3. Stratified Dispersal
12.4. Metapopulation Models

12.1. Random walk
Early population studies concentrated on local population dynamics. However, spatial processes are very important in life-systems of most of the species. They may so significantly modify system behavior that local model would be unable to predict population changes.
Several ecological problems cannot be addressed without analysis of organism dispersal. Examples are: spread of invading species, epidemics, etc.
Let''s take the problem of pest insect control as an example. The first question is what area to treat. If this area is too small it will be immediately colonized by immigrants. Crop rotation is often used to prevent propagation of pests, but the distance between fields with the same crop in two consecutive years should be separated further than migration distance. Finally, many insect pests are sampled using traps (pheromone-baited traps or UV-traps). To determine pest density from trap catches it is important to know dispersal abilities of the insect.
The main problem: how many organisms disperse beyond a specific distance? F1
Random walk is simulated here assuming that 50% individuals stay at the same place, 25% move to the left, and 25% move to the right. After several time steps the distribution of organisms becomes close to the normal distribution F2
Normal (=gaussian) distribution corresponds to equation F3
12.2. Diffusion Models
Advantage of diffusion models is that they can be applied to any initial distribution of organisms. The most simple diffusion model in 1-dimensional space is F4
Model assumptions:
• all individuals simultaneously disperse and reproduce;
• there is no variation in dispersal abilities of individuals.
Skellam''s model predicts that if a population was released at a single point, then its spatial distribution will be a 2-dimensional normal distribution F5
One of the most interesting features of this model is that it predicts the asymptotic rate of expansion of population front. The rate of population expansion, V, is defined as the distance between sites with equal population densities in two successive years F6
Model of Skellam (1951).
Both parameters, r and D, can be estimated in independent experiments. Intrinsic rate of population increase can be determined from the life-table. Diffusion coefficient D can be estimated using mark-recapture experiments. For example, if marked animals are released within a uniform grid of traps, then diffusion coefficient is estimated as F7
12.1. Random walk
Early population studies concentrated on local population dynamics. However, spatial processes are very important in life-systems of most of the species. They may so significantly modify system behavior that local model would be unable to predict population changes.
Several ecological problems cannot be addressed without analysis of organism dispersal. Examples are: spread of invading species, epidemics, etc.
Let''s take the problem of pest insect control as an example. The first question is what area to treat. If this area is too small it will be immediately colonized by immigrants. Crop rotation is often used to prevent propagation of pests, but the distance between fields with the same crop in two consecutive years should be separated further than migration distance. Finally, many insect pests are sampled using traps (pheromone-baited traps or UV-traps). To determine pest density from trap catches it is important to know dispersal abilities of the insect.
The main problem: how many organisms disperse beyond a specific distance? F1
Random walk is simulated here assuming that 50% individuals stay at the same place, 25% move to the left, and 25% move to the right. After several time steps the distribution of organisms becomes close to the normal distribution F2
Normal (=gaussian) distribution corresponds to equation F3
12.2. Diffusion Models
Advantage of diffusion models is that they can be applied to any initial distribution of organisms. The most simple diffusion model in 1-dimensional space is F4
Model assumptions:
• all individuals simultaneously disperse and reproduce;
• there is no variation in dispersal abilities of individuals.
Skellam''s model predicts that if a population was released at a single point, then its spatial distribution will be a 2-dimensional normal distribution F5
One of the most interesting features of this model is that it predicts the asymptotic rate of expansion of population front. The rate of population expansion, V, is defined as the distance between sites with equal population densities in two successive years F6
Model of Skellam (1951).
Both parameters, r and D, can be estimated in independent experiments. Intrinsic rate of population increase can be determined from the life-table. Diffusion coefficient D can be estimated using mark-recapture experiments. For example, if marked animals are released within a uniform grid of traps, then diffusion coefficient is estimated as F7
Example:
The muskrat (Ondatra zibethica) was introduced to Europe in 1905 near Prague. Since that time its area expanded, and the front moved with the rate ranging from 0.9 to 25.4 km/yr. Intrinsic rate of population increase was estimated as 0.2-1.1 per year, and diffusion coefficient ranged from 51 to 230 sq.km/yr. Predicted spread rate (6.4-31.8 km/yr) corresponds well to actual rates of spread.
12.3. Stratified Dispersal
One of the major limitations of diffusion models is the assumption of continuous spread. In nature many organisms can move or can be transferred over large distances. If spread was continuous, then islands would never be colonized by any species. Discontinuous dispersal may result in establishment of isolated colonies far away from the source population.
Passive transportation mechanisms are most important for discontinuous dispersal. They include wind-borne transfer of small organisms (especially, spores of fungi, small insects, mites); transportation of organisms on human vehicles and boats. Discontinuous long-distance dispersal usually occurs in combination with short-distance continuous dispersal. This combination of long- and short-distance dispersal mechanisms is known as stratified dispersal (Hengeveld 1989).
Stratified dispersal includes:
• establishment of new colonies far from the moving population front;
• growth of individual colonies;
• colony coalescence that contributes to the advance of population front
The area near the advancing population front of the pest species can be subdivided into 3 zones:
• Uninfested zone where pest species is generally absent
• Transition zone where isolated colonies become established and grow
• Infested zone where colonies coalesced
A good example of a population with stratified dispersal is gypsy moth. Gypsy moth egg masses can be transported hundred miles away on human vehicles (campers, etc.). New-hatched larvae become air-borne and can be transferred to near-by forest stands. Distribution of gypsy moth counts in pheromone traps in the Appalachian Mts. (Virginia & West Virginia) in 1995 is shown in the right figure. Isolated populations are clearly visible. The US Forest Service Slow-the-Spread project has a goal to reduce the rate of gypsy moth expansion by detecting and eradicating isolated colonies located just beyond the advancing population front
Metapopulation model of stratified dispersal
Sharov and Liebhold (1998, Ecol. Appl. 8: 1170-1179. [Get a PDF reprint!]) have developed a metapopulation model of stratified dispersal. This model is based on two functions: colony establishment rate and colony growth rate. The probability of new colony establishment, b(x), decreases with distance from the moving population front, x F8
The population front is defined as the farthest point where the average density of individuals per unit area, N, reaches the carrying capacity, K:
N = K.
The rate of spread, v, can be determined using the traveling wave equation. We assume that the velocity of population spread, v, is stationary. Then, the density of colonies per unit area m(a,x) of age a at distance x from the population front is equal to colony establishment rate a time units ago. At that time, the distance from the population front was x + av. Thus, m(a,x) = b(x + av). The average numbers of individuals per unit area at distance x from the population front is equal to F9
The population of each colony increases exponentially
n(a) = noexp(ra)
After substituting these functions b(x) and n(a) into the traveling wave equation, we get the following equation F10
This model can be used to predict how barrier zones (where isolated colonies are detected and eradicated) reduce the rate of population spread. We will assume that the barrier zone is placed in the transition zone at some particular distance from the population front. Because new colonies are eradicated in the barrier zone, we can set the colony establishment rate, b(x), equal to zero within the barrier zone as it is shown in the figure below F11
If this new function b(x) is used with the traveling wave equation, we get the rate of spread with the barrier zone. The figure below shows the effect of barrier zone on the rate of population spread. Relative width of the barrier zone is measured as its proportion from the width of the transition zone. Relative reduction of population spread is measured as 1 minus the ratio of population spread rate with the barrier zone to the maximum rate of population spread (without barrier zone) F12.
Cellular automata models of stratified dispersal
Cellular automata is a grid of cells (usually in a 2-dimensional space), in which each cell is characterized by a particular state. Dynamics of each cell is defined by transition rules which specify the future state of a cell as a function of its previous state and the state of neighboring cells. Traditional cellular automata considered close neighborhood cells only. However, in ecological applications it is convenient to consider more distant neighborhoods within specified distance from the cell F13

Get a print pdf
http://home.comcast.net/~sharov/pdf/bioecon.pdf
http://home.comcast.net/~sharov/pdf/quantit.pdf
http://home.comcast.net/~sharov/pdf/optim.pdf

Next -Metapopulation models
main Text
EXEL- Random walk - Diffusion Model
EXEL - Model of SKELLAM
Model: The Spread of Gypsi Moth