In a paper from 2001, Dambacher and Rossignol made a curious observation: Fibonacci numbers appear in the adjoint and absolute feedback matrices that result from a weighted-predictions matrix type analysis (Dambacher et al. 2003) on food chains. The weighted-predictions matrix analysis is a way of predicting how species in a food web will respond to a the press perturbation of one of the species, so that implies that the pattern of species response to certain kinds of disturbance follows a neat mathematical pattern.

To understand where these Fibonacci numbers come from, a paper by Usmani (1994) is useful. First some background:-

Background

If $$D_n$$ is an $$n \times n$$ tridiagonal matrix,

$D_n = \begin{bmatrix} a_1 &b_1 & 0 & \ldots & 0 \\ c_1 & a_2 &b_2 & & \vdots \\ 0 & c_2 &\ddots&\ddots& 0 \\ \vdots & &\ddots&\ddots& b_{n-1} \\ 0 & \dots & 0 & c_{n-1} & a_n \\ \end{bmatrix},$

then a recurrence relationship for the determinant of $$D_n$$ can be found taking cofactors of the last row and then cofactors of the last column to give

$\tag{1} \label{1} |D_n| = a_n |D_{n-1}| - b_{n-1} c_{n-1} |D_{n-2}|.$

The inverse of $$D$$ can be found by

$\tag{2} \label{2} (D^{-1})_{ij} = \begin{cases} (-1)^{i+j} b_i \ldots b_{j-1} \theta_{i-1} \phi_{j+1} / \theta_n, & \text{if } i \leq j \\ (-1)^{i+j} c_j \ldots c_{i-1} \theta_{j-1} \phi_{i+1} / \theta_n, & \text{if } i > j \end{cases}$

where

$\tag{3} \label{3} \theta_i = a_i \theta_{i-1} - b_{i-1} c_{i-1} \theta_{i-2},$

with $$\theta_0 = 1$$ and $$\theta_1 = a_1$$, and where

$\tag{4} \label{4} \phi_i = a_i \phi_{i+1} - b_i c_i \phi_{i+2},$

with $$\phi_{n+1} = 1$$ and $$\phi_n = a_n$$.

Food chain

Now let’s consider the case of a simple food chain.

Let $$A_n$$ be a simplified form of the Jacobian resulting from an $$n$$-species chain, where the elements $$-1, 0,$$ and $$+1$$ indicate negative, zero, and positive effect upon species $$i$$ from species $$j$$, respectively. Then $$A_n$$ is a tridiagonal matrix

$\tag{5} \label{5} A_n = \begin{bmatrix} -1 &+1 & 0 & \ldots & 0 \\ -1 & -1 &+1 & & \vdots \\ 0 & -1 &\ddots&\ddots& 0 \\ \vdots & &\ddots&\ddots& +1 \\ 0 & \dots & 0 & -1 & -1 \\ \end{bmatrix}.$

The response of the steady-states of species to small press-perturbations depend upon the signs in the sensitivity matrix $$S$$ (Nakajima 1992)

$\tag{6} \label{6} S = (-A)^{-1} = \frac{\text{adj}(-A)}{|-A|}.$

Given that there are no countervailing feedbacks in a simple trophic chain, then the system has perfect sign-determinacy in the sense used in Dambacher et al. (2003), and

$\tag{7} \label{7} \text{sign}(S_{ij}) = \text{sign}(|-A| \: \text{adj}(-A)).$

Applying Equation \ref{1}, the determinant

$|-A_n| = |-A_{n-1}| + |-A_{n-2}|$

with $$\lvert A\rvert = 1$$ and $$\lvert -A_2\rvert = 2$$, therefore

$\tag{8} \label{8} |-A_n| = F_{n+1}$

where $$F_i$$ denotes the $$i$$th Fibonacci number. Given that the Fibonacci numbers, and therefore the determinants, are positive, we can strengthen Equation \ref{7} to say that

$\tag{9} \label{9} \text{sign}(S_{ij}) = \text{sign}(\text{adj}(-A)).$

Consequently, Equation \ref{9} ensures that the sign of the local response of all species in a chain to a press perturbation can be determined from the adjugate of the signed $$(-1,0,1)$$ Jacobian alone.

Applying the work of Usmani (1994) to $$A_n$$ from Equation \ref{5}, Equation \ref{4} gives

$\phi_{n-1} = \phi_{n} + \phi_{n+1}$

with $$\phi_n = 1$$ and $$\phi_{n+1} = 1$$ so that

$\tag{10} \label{10} \phi_i = F_{n+2-i}.$

Equation \ref{3} gives

$\theta_i = \theta_{i-1} + \theta_{i-2}$

with $$\theta_0 = 1$$ and $$\theta_{1} = 1$$ so that

$\tag{11} \label{11} \theta_i = F_{i+1}.$

Finally substituting Equation \ref{10} and \ref{11} into Equation \ref{2}, rearranging Equation \ref{6} and substituting in Equation \ref{8}

$\text{adj}(-A)_{ij} = |-A| (A^{-1})_{ij} = \begin{cases} (-1)^{2j} F_i F_{n-j+1} & \text{if } i \leq j \\ (-1)^{i+j} F_j F_{n-i+1} & \text{if } i > j \end{cases}$

which describes a matrix with positive elements on and above the diagonal, and alternating signs below the diagonal

$\label{12} \tag{12} \text{adj}(-A)_{ij} = \begin{bmatrix} +F_1 F_n & +F_1 F_{n-1} & \ldots & +F_1 F_{1} \\ -F_1 F_{n-1} & +F_2 F_{n-1} & & +F_2 F_{1} \\ +F_1 F_{n-2} & -F_2 F_{n-2} & & +F_3 F_{1} \\ \vdots & & \ddots & \vdots \\ (-1)^{i+j} F_1 F_1 & (-1)^{i+j} F_2 F_1 & \ldots & +F_n F_{1} \\ \end{bmatrix}$

We can compare Equation \ref{12} to Equation 16 in Dambacher and Rossignol (2001) to see that they are the same thing.

In summary, Usmani’s (1994) paper allows us to see why Fibonacci numbers appear in the adjoint of a food chain’s community matrix, and also shows that species’ responses to press perturbation will have alternating sign down the food chain.

References

Dambacher, J. M., Li, H. W. and Rossignol, P. A. (2003). Qualitative predictions in model ecosystems, Ecological Modelling 161(1): 79–93.

Dambacher, J. M. and Rossignol, P. A. (2001). The golden rule of complementary feedback, ACM SIGSAM Bulletin 35(4): 1–9.

Nakajima, H. (1992). Sensitivity and stability of flow networks, Ecological Modelling 62(1): 123–133.

Usmani, R. A. (1994). Inversion of a tridiagonal Jacobi matrix, Linear Algebra and its Applications 212: 413–414.