M7116 interaktivně

Authors

Lucia Kajanová

…

Štěpán Zapadlo

Published

May 8, 2025

Histogramy

Tady je nějaký text, viz (1).

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Number of bins:

\[ x^2 + \int x \tag{1}\]

Tady

Number of bins:

Caswell two stage model

Mathematical background

The matrix population model

\[ n(t + 1) = A n(t) \]

is essentially a vector linear recurrence relation, or a system of linear autonomous difference equations.. An analytical solution is given by

\[ n(t) = A^t n(0), \]

where \(n(0)\) is the vector of initial conditions. Rewriting this using the Jordan canonical form yields

\[ n(t) = W J^t W^{-1} n(0). \]

From Perron–Frobenius theory, we can derive the asymptotic behavior for certain matrices.
See Caswell (1989, 57)

Primitive Matrices

A primitive matrix \(A\) has a positive eigenvalue \(\lambda_1\) that is strictly greater in modulus than any other eigenvalue. This \(\lambda_1\) is called the dominant eigenvalue of \(A\). The associated right eigenvector \(w_1\) is positive is referred to as dominant. Similarly, \(v_1\) is the positive left eigenvector associated with \(\lambda_1\).

We can write:

\[ n(t) = \begin{pmatrix} w_1 & \widetilde{W} \end{pmatrix} \begin{pmatrix} \lambda_1^t & \mathbf{0}^T \\ \mathbf{0} & \widetilde{J}^t \end{pmatrix} \begin{pmatrix} v_1^T \\ \widetilde{V} \end{pmatrix} n(0) = \lambda_1^t w_1 v_1^T n(0) + \widetilde{W} \widetilde{J}^t \widetilde{V} n(0). \]

For large \(t\), we have:

\[ n(t) \sim c \lambda_1^t w_1, \]

where \(c = v_1^T n(0) > 0\).

The total population size at time \(t\) is given by the 1-norm of the population vector:

\[ N(t) = \| n(t) \|_1 = \sum_{i=1}^k n_i(t) \sim \lambda_1^t c \| w_1 \|_1. \]

Regardless of the initial population structure \(n(0)\), the population size behaves asymptotically like a geometric sequence with ratio \(\lambda_1\), and the population composition is proportional to the components of the eigenvector \(w_1\).

See Caswell (1989, 61)

Imprimitive and Irreducible Matrices

In this case, the matrix \(A\) has eigenvalues:

\[ \lambda_j = \lambda_1 e^{\frac{2(j-1)}{d} \pi i}, \quad j = 1, 2, \dots, d, \]

where \(d\) is the index of imprimitivity of the matrix \(A\), and \(\lambda_1\) is a positive real number. Other eigenvalues satisfy \(\lambda_1 > |\lambda|\).

We derive:

\[ n(t) = \lambda_1^t \sum_{j=1}^d c_j e^{\frac{2(j-1)}{d} t \pi i} w_j + \widetilde{W} \widetilde{J}^t \widetilde{V} n(0), \]

where \(c_j = v_j^T n(0)\). Thus, the solution of the projection equation with an irreducible and imprimitive matrix \(A\) is asymptotically equivalent to the vector sequence: \[ n(t) \sim \lambda_1^t \sum_{j=1}^d c_j e^{\frac{2(j-1)}{d} t \pi i} w_j. \]

Defining

\[ s(t) = \sum_{j=1}^d c_j e^{\frac{2(j-1)}{d} t \pi i} w_j, \]

yields:

\[ n(t) \sim \lambda_1^t s(t). \]

Also,

\[ s(t + d) = s(t), \quad \frac{1}{d} \sum_{l=0}^{d-1} s(t + l) = \frac{c_1}{d} w_1. \]

This result implies that the solution behaves like a geometric sequence with ratio \(\lambda_1\), modulated by a \(d\)-periodic structure.

If all eigenvectors \(w_j\) are normalized, the total population size satisfies:

\[ N(t) \sim \lambda_1^t \left\| \sum_{j=1}^d c_j e^{\frac{2(j-1)}{d} t \pi i} w_j \right\|_1. \]

See Caswell (1989, 63)

Damping ratio

Let \(\mu\) be the largest modulus of the eigenvalues of the \(k \times k\) matrix \(A\) that are smaller than \(\lambda_1\); in the case \(d = k\), where \(d\) is the imprimitivity index, we define \(\mu = 0\).

Then the damping ratio is defined as:

\[ \rho = \frac{\lambda_1}{\mu}, \]

and for \(\mu = 0\) we set \(\rho = \infty\).

The damping ratio expresses the rate of convergence to the stabilized structure. See Caswell (1989, 69)

Sensitivity of the growth rate

The sensitivity of the characteristic \(\chi = \chi(A)\) to the component \(a_{ij}\) of the projection matrix \(A\) is defined as

\[ \frac{\partial \chi}{\partial a_{ij}}. \]

In particular, for the growth rate \(\lambda_1\), we have

\[ \frac{\partial \lambda_1}{\partial a_{ij}} = \frac{v_i w_j}{v^T w}. \]

The sensitivity matrix \(S\) is then defined by

\[ S = (s_{ij}) = \left( \frac{\partial \lambda_1}{\partial a_{ij}} \right) = \left( \frac{v_i w_j}{v^T w} \right) = \frac{1}{v^T w} v w^T. \]

See Caswell (1989, 118)

Elasticity of the growth rate

The elasticity of the characteristic \(\chi = \chi(A)\) with respect to the component \(a_{ij}\) of the projection matrix \(A\) is defined as

\[ \frac{a_{ij}}{\chi} \cdot \frac{\partial \chi}{\partial a_{ij}} = \frac{\partial \ln \chi}{\partial \ln a_{ij}}. \]

In particular, the elasticity \(e_{ij}\) of the growth rate \(\lambda_1\) with respect to the entry \(a_{ij}\) is

\[ e_{ij} = \frac{1}{\lambda_1 v^T w} a_{ij} v_i w_j. \]

The elasticity matrix is given by

\[ E = (e_{ij}) = \frac{1}{\lambda_1 v^T w} A \circ (v w^T), \]

where \(\circ\) denotes the Hadamard (element-wise) product of matrices.

See Caswell (1989, 132)

Long-term behavior of eigenvalues for Caswell’s \(2 \times 2\) matrix model

The long-term behavior of the population vector \(n(t)\), given by the equation

\[ n(t) = A^t n(0), \]

depends on the eigenvalues \(\lambda_i\) of the projection matrix \(A\), as they are raised to higher and higher powers. For this particular model, we have two real eigenvalues. In this situation, we can in some way analyze the behavior of this system given by matrix \(A\) using all of its eigenvalues.
(In more general cases, the contribution of all the smaller eigenvalues to the behavior tends to be less relevant, and the deciding factors are the dominant eigenvalue and whether the matrix is primitive or not, and at least irreducible.)

If \(\lambda_i \in \mathbb{R}\), then:

If \(\lambda_1 > 1\): the population grows exponentially.
If \(\lambda_1 = 1\): the population stabilizes over time.
- If \(0 < \lambda_2 < 1\): the rate of convergence to the stable structure increases as \(\lambda_2\) tends to zero.
- If \(\lambda_2 = 0\): the population stabilizes after just one step.
- If \(-1 < \lambda_2 < 0\): the population stabilizes with damped oscillations.
- If \(\lambda_2 = -1\): the ratio of the components of the solution \(n(t)\) changes periodically, with a period of 2.
If \(0 < \lambda_1 < 1\): the population decays exponentially.

Caswell two stage model

Here is the table for the reader to try some of the aforementioned situations. The first four matrices are primitive, and the last two are imprimitive and irreducible. For all examples, the fertility \(\varphi = 1\) and the initial conditions vector is \(n(0) = (1, 0)^T\).

Example parameter sets to explore with the model
\(\sigma_1\)	\(\sigma_2\)	\(\gamma\)	Matrix \(A\)	\(\lambda_1\)	\(\lambda_2\)	Behavior of population \(\|\|n(t)\|\|_1\)	Damping ratio
\(1\)	\(1\)	\(1\)	\(\pmatrix{0 & 1 \\ 1 & 1}\)	\(\frac {1 + \sqrt{5}} 2\)	\(\frac{1 - \sqrt{5}} 2\)	\(\lambda_1 > 1\) : population grows exponentially	\(\frac{\sqrt{5} - 1} 2\)
\(\frac 2 3\)	\(\frac 2 3\)	\(1\)	\(\pmatrix{\frac 1 3 & 1 \\ \frac 2 3 & 1}\)	\(1\)	\(\frac 1 3\)	\(\lambda_1 = 1\), \(0 < \lambda_2 < 1\): population stabilizes over time	\(3\)
\(\frac 2 3\)	\(\frac 1 3\)	\(\frac 1 2\)	\(\pmatrix{\frac 1 2 & 1 \\ \frac 1 2 & 1}\)	\(1\)	\(0\)	\(\lambda_1 = 1\), \(\lambda_2 = 0\): population stabilizes after one step	\(\infty\)
\(\frac 5 9\)	\(1\)	\(1\)	\(\pmatrix{0 & 1 \\ \frac 5 9 & 1}\)	\(1\)	\(-\frac 5 9\)	\(\lambda_1 = 1\), \(\lambda_2 < 0\): population stabilizes, but exhibits damped oscillations	\(\frac 9 5\)
\(1\)	\(0\)	\(1\)	\(\pmatrix{0 & 1 \\ 1 & 0}\)	\(1\)	\(-1\)	\(\lambda_1 = 1\), \(\lambda_2 = -1\): population is stable, but its structure oscillates with period \(2\)	\(\infty\)
\(\frac 2 3\)	\(0\)	\(1\)	\(\pmatrix{0 & 1 \\ \frac 2 3 & 0}\)	\(\frac {\sqrt 2} 3\)	\(-\frac {\sqrt 2} 3\)	\(\|\lambda_1\| < 1\), \(\lambda_2 = \overline{\lambda_1}\): population decreases, but its structure oscillates with period \(2\)	\(\infty\)

Survival of juveniles (σ₁)

Transition to adults (γ)

Survival of adults (σ₂)

Fertility (f)

Length of projection (T)

Initial number of juveniles (x₁)

Initial number of adults (x₂)

Bibliography

Caswell, Hal. 1989. Matrix Population Models. Sunderland, MA: Sinauer Associates.

Pospíšil, Zdeněk. 2015. “Maticové Populační Modely.” Edited by KOMENDA HOLČÍK Jiří. Brno: elektronická verze "online"; Masarykova univerzita. http://portal.matematickabiologie.cz/index.php?pg=analyza-a-modelovani-dynamickych-biologickych-dat--diskretni-deterministicke-modely.