The Growth Rates of Population Projection Matrix Models in Random Environments

Tatsuro Akamine* and the late Maki Suda

National Research Institute of Fisheries Science, Fisheries Research Agency, Fukuura 2-12-4, Kanazawa, Yokohama 236-8648, Japan

Abstract

Population projection matrix models in random environments are random walk models. The growth rate of the mean population size, which is equal to the maximum eigenvalue of the mean matrix, is better than the average of the intrinsic rates of natural increase calculated by computer simulations, because the population size is more important than the growth rate. The arithmetic mean of the maximum eigenvalues of matrices for all permutations converges to the maximum eigenvalue of the mean matrix. The periodicity of environments is more important than the correlation between environments. Simple matrices and three numerical models are used as examples.

Keywords

eigenvalue, growth rate, Lefkovitch, Leslie, permutation, projection matrix


Received on June 1, 2011

Accepted on August 3, 2011

Online published on October 15, 2011

*Corresponding author at:

e-mail: akabe@affrc.go.jp


1. Introduction

Population projection matrix models, which are called the Leslie matrix model and the Lefkovitch matrix model, have been expanded for random environments (Caswell 2001). These models have already been used in fish population dynamics (Cohen et al. 1983; Quinn and Deriso 1999; Akamine 2009). However, there are two problems with these models. One is the method of estimating the population growth rate and the other is the negative correlation of the successive environmental states. In this paper, we will discuss these problems using simple models.

Akamine (2011) proved the theorem that the arithmetic mean of the maximum eigenvalues of matrices for all permutations of the random walk matrix model converges to the maximum eigenvalue of the mean matrix, which is defined as the population growth rate. Akamine (2010a, b) calculated the maximum eigenvalues of all permutations to estimate their distribution, which is more basic than random simulations generated by computers. We will also discuss these matters.

page top


2. Scalar model

The basic model is

where N is the population size, λ is the growth rate and t is discrete time. Thus, we obtain

Suppose the value of λ(i) is α or β at random. The arithmetic mean of these is

The t-th power is

The right-hand side has 2t permutations of α and β and the left-hand side is an arithmetic mean of all permutations on the right-hand side. This is an exponential model and δ is the mean growth rate. When α = 2, β = 0.5, we obtain δ = 1.25 > 1. Thus, the mean population size will increase.

On the other hand, the logarithm of Eq. (1) is

where r = logλ is the intrinsic rate of natural increase. We then obtain

This is a random walk model. In the above case, the mean logarithm of the population size will not increase because the arithmetic mean of logα and logβ is 0. When t → ∞, the distribution of ∑r approaches to the normal distribution. Thus, the distribution of ∏λ in Eq. (2) approaches to the lognormal distribution.

The definition of the lognormal distribution is

The following are well-established:

where E(…) is the expected value. Although Eq. (7) shows that the mean of logN is μ, Eq. (8) shows that the logarithm of E(N) is μ + σ2/2 > μ.

Which is better for fish population dynamics, Eq. (1) or Eq. (5)? We think Eq. (1) is better than Eq. (5). This problem will be discussed again in Section 8. Therefore, in matrix models, it is better to choose the maximum eigenvalue of the mean matrix for the population growth rate, not the average of the population growth rate. These are described in the following section.

page top


3. Matrix model

The population projection matrix model in quadratic form is defined as follows:

or

where ni is the number of individuals in age or category class of i, a and b are reproduction rates and c and d are survival rates. All constants are not negative. Thus, the projection matrix L is a non-negative matrix.

When this model is modified as

the t-th power is

where λ1 and λ2 are eigenvalues and S is a matrix of eigenvectors. Eigenvalues are roots of the eigenequation

When ν = |λ2|/λ1 < 1, we obtain νt → 0 (t → ∞),

Therefore, the maximum eigenvalue λ1 is defined as the population growth rate. The rank of this matrix is

Basic model (12) is expanded in random environments as

Caswell (2001) defined the population size as

where w0 and w1 are weights. He also defined the average growth rate as

This is expressed exactly as

It is not possible to obtain this value analytically, this is an average of computer simulations.

On the other hand, Cohen et al. (1983) defined the growth rate of the expected population size as

This is equal to

They used a general model

where c(t) is a random variable. They also showed the following equation

and proved

where the right-hand side is the maximum eigenvalue of the mean matrix. Akamine (2011) proved this theorem by using eigenvalues for the random walk matrix model, which is explained in the following section.

page top


4. Theorem

Two environments expressed as P or Q occur at random. The mean matrix of these is

and the t-th power is

where Perm(P, Q, t, i) is a permutation of P and Q whose length is t (for example, PPQPQQQPP when t = 9) and ∑ means the sum of all permutations. This equation can be rewritten as

or

Akamine (2011) proved the equation

which means that the arithmetic mean of the maximum eigenvalues of matrices for all permutations on the random walk matrix model converges to the maximum eigenvalue of the mean matrix.

page top


5. Proof

(Lemma)

The general theory of eigenvalues is as follows: Let

Then

This equation implies

(Proof)

In Eq. (28), the following equations hold:

When the following equation holds,

Eq. (31) will hold because of the above lemma. The linear mapping of P or Q in Eq. (11) projects the basic vectors of the x- and y-axes

to

The inner product of these vectors is

Thus, when ab + cd ≠ 0, the cosine of these vectors is

and the angle of these vectors is |θ| < π/2. When t → ∞, these projected vectors will overlap with each other, because Perm(P, Q, 2t, i) involves P or Q over or equal to t times. Thus, for any Perm(P, Q, t, i), rank[Perm(P, Q, t, i)] → 1 (t → ∞). (Q.E.D.) It is easy to expand this proof for the m-dimensional matrix.

When ab + cd = 0, we can show the exception (A-model, Akamine 2010a, b):

The mean matrix of these is

This matrix projects basic vectors to

These vectors intersect orthogonally and the rank of matrix D is still 2 when t → ∞.

page top


6. Correlation and periodicity

In this section we will discuss the correlation and periodicity of successive environmental states. Let us consider the A-model (Eq. (41)). The squared matrix of D is

In this model, the correlation coefficient of two environmental states, B and C , is ρ = –1.

Let us consider the following matrices:

The mean matrix of these is

Thus, we obtain

We think that the correlation coefficient is difficult to define for this model because there are three environmental states. For these multi-state models, the periodicity of environmental states is more important than the correlation of them.

page top


7. Example

We show three examples for the distributions of eigenvalues as Matsuda and Iwasa (1993)'s M-model:

Tuljapurkar (1989)'s T-model:

and Caswell (2001)'s C-model:

Tables 13 show λ1, λ2 and λ21 when t = 1, ..., 6 for each model. Figure 1 shows the convergence of the arithmetic mean of the eigenvalues and Figs. 24 show the distribution of the eigenvalues and their logarithms when t = 6.

Table 1. Maximum eigenvalues of matrices for permutations in the M-model. Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography.

1)Number of permutations that have the same eigenvalues. 2)Arithmetic mean.

page top


Table 2. Maximum eigenvalues of matrices for permutations in the T-model. Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography.

1)Number of permutations that have the same eigenvalues. 2)Arithmetic mean.

page top


Table 3. Maximum eigenvalues of matrices for permutations in the C-model. Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography.

1)Number of permutations that have the same eigenvalues. 2)Arithmetic mean.

page top


Fig. 1. Convergence of arithmetic means of the maximum eigenvalues of matrices for all permutations to the maximum eigenvalue of the mean matrix. Relative error is tm1(R). Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography.

page top


Fig. 2. Distribution of the maximum eigenvalues of matrices for all permutations and their logarithms in the M-model when t = 6. Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography.

page top


Fig. 3. Distribution of the maximum eigenvalues of matrices for all permutations and their logarithms in the T-model when t = 6. Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography.

page top


Fig. 4. Distribution of the maximum eigenvalues of matrices for all permutations and their logarithms in the C-model when t = 6. Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography.

page top


In these calculations, the following formula is used

This is proved easily as follows (Yano 1974): The left-hand side is

Thus, we can obtain

This is the right-hand side of Eq. (52).

Tuljapurkar (1989) showed that rs = logλs = 0.1954 calculated by computer simulations for the T-model. Table 2 gives us the arithmetic mean of logλ1, which is 0.1974 when t = 6. This is evidence that the distribution of logλ1 approaches to the normal distribution when t → ∞. The T- and C-models are approximately equal to the A-model, which is the reason why they are cautionary (Caswell 2001).

page top


8. Discussion

The mean matrix is defined in general as

Thus, theorem (31) holds in general. Caswell (2001) showed the following relation when t → ∞,

Therefore, N(t) distributes lognormally and

He considered that the intrinsic rate of natural increase rs = logλs is better than logΛ for the population growth rate. However, we think it is not optimal to compare these logarithms. We had better compare logλs with Λ, which is equivalent to comparing logN with N for the population size. We consider that N is better than logN due to two reasons. One is that we use the population size N in fisheries management, not the logarithm population size logN. The other is that the analysis for raw data is better than the analysis for transformed data in statistics. If there is no advantage for raw data, the lognormal distribution is not necessary for data analysis. Therefore, Λ is better than logλs for population projection matrix models.

Cautionary T- and C-models have very large eigenvalues in Tables 2, 3. When these models approximate the A-model, their large values become very important. Thus, we had better estimate the distribution of the growth rate λ1 as a histogram. Calculating λ1 for all permutations in the short-term is more basic than many computer simulations carried out over the long-term in the random walk matrix model. In general projection population matrix models, the vector n(t) has much more information than the scalar N(t) for the population size.

Caswell (2001) showed that the negative correlation between environments is important for good population growth in the C-model, which is approximately equal to the A-model. However, we discussed that the periodicity is more important than the correlation in many environmental states, for which we expanded the A-model to a 3-dimesional model in Section 6.

page top


9. Conclusion

In population projection matrix models, the distribution of the maximum eigenvalues of all permutations approaches to the lognormal distribution. Their arithmetic mean converges to the maximum eigenvalue of the mean matrix. Thus, it is relevant that the population growth rate be defined as the maximum eigenvalue of the mean matrix, not the average of growth rates calculated by computer simulations.

The negative correlation between environments is not so important in many conditional environments. We consider that the periodicity is more important than the correlation.

Acknowledgments

The authors thank Dr. Kazuhiko Hiramatsu of the University of Tokyo, Dr. Ichiro Oohara of the National Research Institute of Fisheries Science and the two anonymous referees for their useful comments on earlier drafts of the manuscript. The authors also thank Ms. Tomoko Yamakoshi for her help with our work.

page top


References

Akamine T. Non-linear and graphical methods for fish stock analysis with statistical modeling. Aqua-BioSci. Monogr. 2009; 2(3): 1–45.

Akamine T. Suisan Sigen no Data Kaiseki Nyuumon. Kouseisha Kouseikaku, Tokyo. 2010a; 178 pp.

Akamine T. Mathematical study of matrix models for fish population dynamics in random environments. Bull. Jpn. Soc. Fish. Oceanogr. 2010b; 74: 208–213 (in Japanese).

Akamine T. Proof of a theorem for the growth rate in matrix models of fish population dynamics. Bull. Jpn. Soc. Fish. Oceanogr. 2011; 75: 29–31 (in Japanese).

Caswell H. Matrix Population Models, 2nd ed. Sinauer Associates, Sunderland. 2001; 722 pp.

Cohen JE, Christensen SW, Goodyear CP. A stochastic age-structured population model of striped bass (Morone saxatilis) in the Potomac river. Can. J. Fish. Aquat. Sci. 1983; 40: 2170–2183.

Matsuda H, Iwasa Y. Nani ga zetsumetsu wo motarasunoka?: hozen seitaigaku no riron. Kotaigun seitaigakkai kaihou. 1993; 50: 2–9.

Quinn TJ, Deriso RB. Quantitative Fish Dynamics. Oxford Univ. Press, New York. 1999; 542 pp.

Tuljapurkar S. An uncertain life: demography in random environments. Theor. Popul. Biol. 1989; 35: 227–294.

Yano K. Senkei Daisuu, Bekutoru to Gyouretsu. Nippon-Hyoron-Sha, Tokyo. 1974; 177 pp.

page top


List of Figures

Fig. 1. Convergence of arithmetic means of the maximum eigenvalues of matrices for all permutations to the maximum eigenvalue of the mean matrix. Relative error is tm1(R). Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography.

Fig. 2. Distribution of the maximum eigenvalues of matrices for all permutations and their logarithms in the M-model when t = 6. Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography.

Fig. 3. Distribution of the maximum eigenvalues of matrices for all permutations and their logarithms in the T-model when t = 6. Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography.

Fig. 4. Distribution of the maximum eigenvalues of matrices for all permutations and their logarithms in the C-model when t = 6. Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography.

Table 1. Maximum eigenvalues of matrices for permutations in the M-model. Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography. 1)Number of permutations that have the same eigenvalues. 2)Arithmetic mean.

Table 2. Maximum eigenvalues of matrices for permutations in the T-model. Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography. 1)Number of permutations that have the same eigenvalues. 2)Arithmetic mean.

Table 3. Maximum eigenvalues of matrices for permutations in the C-model. Modified from Bull. Jpn. Soc. Fish. Oceanogr., 74, Akamine, Mathematical study of matrix models for fish population dynamics in random environments, 208–213, © 2010, with permission from the Japanese Society of Fisheries Oceanography. 1)Number of permutations that have the same eigenvalues. 2)Arithmetic mean.

page top