Biological meaning of Lotka-Volterra Jacobian matrices

Question

In a book about community ecology, I learned about Lotka-Volterra models and dynamic food web models (I am not an ecologist in the first place).

In one of the chapters, a Jacobian matrix of partial derivatives of the Lotka-Volterra model at equilibrium densities is given: \begin{bmatrix}-&-&0&0\\+&0&-&0\\0&+&0&-\\0&0&+&0\end{bmatrix} where + and – represent positive or negative values.

This matrix is later interpreted in terms of direction of trophic links in a food web, but the reasoning allowing to jump form one to the other is not detailed.
So what is the biological meaning of these values?

Answer 1

The Jacobian tells how the system changes along different state variables (which can be, for instance, the concentrations of the predator and the prey).

The Jacobian matrix by itself doesn't give you a lot of intuitive information. However, the eigenvalues of the Jacobian matrix at the equilibrium point tell you the nature of the steady state. For example if all the eigenvalues are negative then the system is stable at that point and the equilibrium point is called a stable node. If some of them are positive and some are negative then the system is metastable at that point and the point is called a saddle point. If all eigenvalues are positive then the point is unstable. If the eigenvalues have an imaginary part then the system exhibits oscillations. All these properties are local i.e. they just tell you the behaviour of the system near the equilibrium point.

I hope I am clear. This is not in a strict sense, a biological question. If you want to know more then read a book on nonlinear dynamics. Non-linear dynamics and Chaos by Steven Strogatz is a good place to start.

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EDIT

When the system is locally linear (linearized at a given point) then the system dynamics at that point can be described as $$\frac{dx}{dt}=J.x$$

Where $x$ is the state vector (a vector of the variables in the system) and $J$ is the Jacobian matrix. In this situation, the signs will denote the effect of a variable on the other variable. However, I am not sure (I feel it is highly unlikely) if this can be used to make any inferences about the food web.

Answer 2

The matrix above depicts the signs of the partial change of a function with respect to each species. Because there are 4 rows and columns, this means there are 4 species. Each row represents a partial change in each of the 4 differential equations representing the rates of change for each species with respect to each species, which is each column. To rewrite the Jacobian for species 1–4 ($N_1, N_2, N_3, N_4)$: $$\mathbf{J} = \left[\begin{smallmatrix} \frac{\partial F_1}{\partial N_1} & \frac{\partial F_1}{\partial N_2} & \frac{\partial F_1}{\partial N_3} &\frac{\partial F_1}{\partial N_4} \\ \frac{\partial F_2}{\partial N_1} & \frac{\partial F_2}{\partial N_2} & \frac{\partial F_2}{\partial N_3} &\frac{\partial F_2}{\partial N_4} \\ \frac{\partial F_3}{\partial N_1} & \frac{\partial F_3}{\partial N_2} & \frac{\partial F_3}{\partial N_3} &\frac{\partial F_3}{\partial N_4} \\ \frac{\partial F_4}{\partial N_1} & \frac{\partial F_4}{\partial N_2} & \frac{\partial F_4}{\partial N_3} &\frac{\partial F_4}{\partial N_4} \\ \end{smallmatrix}\right] $$ The signs above correspond the effect of species on each other's rates of change. For a first example, the first row and column is effect of species 1 on its own rate of change. It's negative, meaning that it's rate of change decreases with increasing density (crowding). A second example is the first row and second column—species 2 has a negative effect on the growth of species 1. In a food web context, this means that species 2 consumes species 1. A third example is the first row and third column—there is no effect of species 3 on species 1. Lastly, the second row and first column has a positive sign, meaning that species 1 has a positive effect on the growth rate of species two. In a food web context, that means that species 2 consumes species 1.

Notice that in food webs, for each positive effect, there is a corresponding negative effect and vice versa (i.e., species grow from eating another species, and species shrink from being eaten).

In sum, (1) species 1 is the only species who is affecting by crowding, (2) species 1 is consumed by species 2, (3) species 2 is consumed by species 3, (4) species 3 is consumed by species 4, and (5) no other species interact.