doi:10.5047/absm.2011.00403.0095
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National Research Institute of Fisheries Science, Fisheries Research Agency, Fukuura 2-12-4, Kanazawa, Yokohama 236-8648, Japan
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.
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
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.
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 2^{t} 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 + σ^{2}/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.
The population projection matrix model in quadratic form is defined as follows:
or where n_{i} 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 λ_{1} and λ_{2} are eigenvalues and S is a matrix of eigenvectors. Eigenvalues are roots of the eigenequation When ν = |λ_{2}|/λ_{1} < 1, we obtain ν^{t} → 0 (t → ∞), Therefore, the maximum eigenvalue λ_{1} is defined as the population growth rate. The rank of this matrix isBasic model (12) is expanded in random environments as
Caswell (2001) defined the population size as where w_{0} and w_{1} 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.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.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 → ∞.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.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 1–3 show λ_{1}, λ_{2} and λ_{2}/λ_{1} when t = 1, ..., 6 for each model. Figure 1 shows the convergence of the arithmetic mean of the eigenvalues and Figs. 2–4 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.
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.
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 ^{t}√m/λ_{1}(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.
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 r_{s} = logλ_{s} = 0.1954 calculated by computer simulations for the T-model. Table 2 gives us the arithmetic mean of logλ_{1}, which is 0.1974 when t = 6. This is evidence that the distribution of logλ_{1} 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).
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 r_{s} = logλ_{s} is better than logΛ for the population growth rate. However, we think it is not optimal to compare these logarithms. We had better compare logλ_{s} 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 logλ_{s} 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 λ_{1} as a histogram. Calculating λ_{1} 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.
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.
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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 ^{t}√m/λ_{1}(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.