 | Model of Leslie |
F1- the equation [1]. |
F2- the equation [2]. |
F3, there Nx is the vector of age distribution in the population at time t, and A is the transition matrix |
F4, When a matrix is multiplied by a vector, we take the 1st row of the matrix, multiply each number by the corresponding number in the vector-column and then sum all products. This sum is the value of the 1st element in the result vector. Then we take the 2nd row of the matrix, multiply it by the same vector and the sum becomes the 2nd element in the result vector. In the same way we can estimate all elements of the result vector. The first element of the result vector corresponds to the equation [2], and all other elements correspond to the equation [1]. |
F5 - In this case, the sum of matrix elements in each column equals to 1 because it is assumed that each system passes through a series of states and neither die nor reproduce |
F6. - This equation can be used to simulate as many time steps as necessary |
F7- The first graph shows exponential population growth (it becomes linear in a log scale) after several initial years. |
F7 - The second graph shows convergence of age distribution to a stable age distribution |
F8 |
F9. |
F10 - equation [3], where |
Now, equation [2] of the Leslie model can be re-written as |
the equation [4]. After initial damped fluctuations, the Leslie model shows exponential growth and age distribution stabilizes. Thus, the number of organisms in any age class will grow exponentially. In particular, the number of new-born organisms increases exponentially. |
then |
the equation [5] |
Now we combine equations [4] and [5] |
The term N0,t-x can be taken out of the summation expression and then we get the equation 6 |
F11. |
|
Equation [5] can be re-written as F12 |
F13 |
equation [7] |
Transition matrix can be modified to incorporate these features. This matrix has non-zero diagonal elements, and thus, some proportion of organisms remain in the same age class when time increases. As a result, development goes slower than it would be in the original Leslie model. The rate of development can be adjusted by changing relative values of diagonal and sub-diagonal elements |
Transition matrix can be modified to incorporate these features. This matrix allows organisms to leap over several age intervals in one time step. As a result, development goes faster than it would be in the original Leslie model. The rate of development can be adjusted by changing the length of age leaps. |
Partitioning the life cycle into stages.Many invertebrate species have a complex life cycle that includes several stages. For example, holometabolous insects usually have 4 stages: egg, larvae, pupae, and adult. Each of these stages may include several age intervals: In these models, age is no longer measured in calendar time units (e.g., days or years). Instead, it is measured in independent units which can be interpreted as "physiological age". The concept of physiological age will be discussed in details in the next chapter. It can be used to define "rate of development" as the average increment of physiological age per calendar time unit. |
Model of Leslie
The model of Leslie is one of the most healivy used models in population ecology. This is a discrete-time model of an age-structured population which describes development, mortality, and reproduction of organisms. The model is formulated using linear algebra. This model is mostly used to answer the following two questions:
What is the rate of exponential growth (intrinsic rate of increase)?
What is the proportion of each age class in the stable age distribution?
7.1. Model structure
7.2. Model behavior
7.3. Intrinsic rate of population increase
7.4. Stable age distribution
7.5. Modifications of the Leslie model
7.1. Model Structure
The model of Leslie (1945) describes 3 kinds of ecological processes:
Development (progress through the life cycle)
Age-specific mortality
Age-specific reproduction
Variables and parameters of the model:
Nx,t = number of organisms in age x at time t (age is measured in the same units as time t). Usually, only females are considered and males are ignored because, as a rule, the number of males does not affect population growth.
sx = survival of organisms in age interval from x to x+1.
mx = average number of female offsprings produced by 1 female in age interval from x to x+1 (mortality of parent and/or offspring organisms is included)
There are two equations f1 and F2.
Equation [1] represents development and mortality, whereas equation [2] represents reproduction. Equation [2] specifies the number of individuals in the first age class and equation [1] specifies the number of individuals in all other age classes. In the equation [1], the number of individuals in age x+1 in time t+1 equals to the number of individuals in the previous age and previous time multiplied by age-specific survival rate sx. In the equation [2] the number of new-born organisms equals to the number of mothers (Nx,t) multiplied by the numbers of offspring produced (mx). The number of offsprings is summed over all ages of mothers.
These two equations can be combined into one matrix equation F3.How to read matrix models?
Each column specifies the fate of organisms in specific state. The number in the intersection of column i and row j indicates how many organisms in state j are produced by one organism in state i. In the Leslie model, organisms'' state is defined by age only. For example, the third column corresponds to age a=2. An organism in age 2 produces m offsprings of age 0 (first cell in the column), and goes to age class 3 with probability s (the cell under main diagonal).
Most simple stochastic process is defined by a matrix of transition probabilities between states F5.
Leslie model is more complex because the sum of matrix elements in each column is not necessary equal to 1. This is a "branching process" because the life trajectory of a parent branches into life trajectories of its offsprings.
Matrix models are easy to iterate in time. In the next time step we again multiply the transition matrix by the vector of age distribution F6.
7.2. Model Behavior
Two major characteristics of the behavior of Leslie model:
A few damping oscillations are followed by an exponential growth
Age distribution approaches a stable age distribution
These 2 features can be seen in the graphs that show simulations of sheep population dynamics F7. In this simulation we started from a population of 100 new-born sheep F8.Here is an example of the model of Leslie implemented using Excel spreadsheet.
7.3. Estimation of the intrinsic rate of population increase
Previously we discussed an approximate equation F9.
The model of Leslie gives an accurate estimation of r.
Method #1
For simplicity we take years as time units. However, the same logic can be applied to days or weeks.
The number of organisms in age x in year t is equal to the number of new-born organisms (x=0) x years ago multiplied by their survival (lx) till age x- F10.
Equation [6] can be used to estimate r. The sum at the left side can be estimated for different values of r, and then we can select the r-value that makes this sum equal to 1. It makes sense to start with the r-value estimated using the approximate method discussed in the previous chapter. Thus, we start with r=0.181 and get the sum [6] equal to 0.9033. When r increases, then the value of the sum [6] decreases because r is included as a negative exponent. Obtained value of the sum appeared to be less than 1, and thus, we need to try smaller values of r. Let''s select r=0.16. Then the sum is equal to 1.0092. The exact value of r can be found by linear interpolation F11.
Now we check the solution: when r=0.1617 then the sum [6] is equal to 1.00014 which is very close to 1.
Note: In Pielou (1978), this example is estimated in a different way which is difficult to understand. She constructed a different matrix by adjusting reproduction rates and provided no explanation for this adjustment.
Method #2
Intrinsic rate of population increase can be estimated as the logarithm of the only real and positive eigenvalue of the transition matrix. The theory of eigenvalues is the central topic in linear algebra. It is used to reduce multidimensional problems to one-dimensional problems. I recommend to study this topic for those students who plan to be quantitative ecologists. Here we will only estimate the eigenvalue using available software without going into details of the algorithm. The only real and positive eigenvalue of our matrix is equal to =1.176. Then, r = ln() = 0.162 which is very close to the value estimated by method #1.
Checking the result
Obtained value of r can be checked by estimating the regression of log population numbers versus time. Initial years should be ignored because age structure has not been stabilized yet. The slope of this regression should be equal to r. If we take the time interval from t = 25 to 50, then the regression equation is ln(N) = 4.3557 + 0.1617 t. Regression slope is exactly equal to r estimated by method #1.
.4. Estimation of stable age distribution
Equation [5] can be re-written as F12.
Substituting this equation into [3] we get the relationship between the number of organisms in age x and in age 0 in a stable age distribution F13.
Now we can estimate the proportion of organisms, c , in age x (equation [7]).
Age,
x lx exp(-rx) lxexp(-rx) cx Simulated cx
0 1.000 1.0000 1.0000 0.2413 0.2413
1 0.845 0.8507 0.7188 0.1734 0.1734
2 0.824 0.7237 0.5963 0.1439 0.1439
3 0.795 0.6156 0.4894 0.1181 0.1181
4 0.755 0.5237 0.3954 0.0954 0.0954
5 0.699 0.4455 0.3114 0.0751 0.0751
6 0.626 0.3790 0.2373 0.0572 0.0572
7 0.532 0.3224 0.1715 0.0414 0.0414
8 0.418 0.2743 0.1147 0.0277 0.0277
9 0.289 0.2333 0.0674 0.0163 0.0163
10 0.162 0.1985 0.0322 0.0078 0.0078
11 0.060 0.1689 0.0101 0.0024 0.0024
Total 4.1445 1.0000 1.0000
Age distribution estimated using equation [7] (column 5) coincided with simulated age distribution after 50 iterations of the model.
7.5. Modifications of the Leslie model
1. Variable matrix elements. Survival and reproduction rate of organisms may depend on a variety of factors: temperature, habitat characteristics, natural enemies, food, etc. To represent these dependencies, the elements of Leslie model can be replaced by equations that specify survival and reproduction rates as functions of various factors. Equations can be obtained from experimental data.
2. Distributed delays. Age and time are equivalent in the original Leslie model, and thus, all organisms develop synchronously with constant rate. However, development rate of invertebrates and plants is not constant: it depends on temperature and may vary among individuals. Individual variation of development rates is called "distributed delay" because there is a distribution of time when organisms reach maturity. Transition matrix can be modified to incorporate these features.
3. Partitioning the life cycle into stages - f14.
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Main text
Excel spreadsheet "leslie.xls"
Linear and matrix algebra
