Leslie matrix with no dominant eigenvalue

[Darkmisc]

Homework Statement

What is the consequence for a population model if the Leslie matrix used has no dominant eigenvalue?

Homework Equations

x(k) = Ax(k-1), where A is the Leslie matrix, x is a vector representing the initial population distribution.

x(k) is the vector of the population distribution for time period k.

The Attempt at a Solution

All I understand of the dominant eigenvalue is that over long time periods (mutliple iterations of x(k) = Ax(k-1) ), x(k) approaches lambda*x(k-1). In other words, the population distribution tends towards that given by the eigenvector associated with the dominant eigenvalue.

I understand the proof of this http://online.redwoods.cc.ca.us/instruct/darnold/LinAlg/leslie2/context-leslie2-p.pdf

which uses the diagonalisation of A.

I've read that one consequence of A not having a dominant eigenvalue is that the population numbers will rise and fall in waves. Is this the only consequence? Can anyone direct me to resources on this? All I seem to find are pages about matrices with dominant eigenvalues, rather than ones with no dominant eigenvalue.

 

[mighty2000]

Dear forum post,

Thank you for your question regarding the consequences of a population model having no dominant eigenvalue in the Leslie matrix used. I can provide some insight into this topic.

Firstly, let's define what a dominant eigenvalue is and its significance in a Leslie matrix. The dominant eigenvalue is the largest eigenvalue of the matrix, and it corresponds to the growth rate of the population. In other words, it tells us how quickly the population is growing or declining. If there is no dominant eigenvalue, it means that there is no single growth rate that dominates the population dynamics. This can have several consequences on the population model.

One consequence is that the population numbers will fluctuate in a non-predictable manner. This is because the growth rates of the different age groups in the population are not synchronized, leading to unpredictable fluctuations in the overall population size. This can be seen as the population rising and falling in waves, as you mentioned in your post.

Another consequence is that the population may not reach a steady state. In a population model with a dominant eigenvalue, the population will eventually reach a stable equilibrium, where the population size remains constant over time. However, in a model with no dominant eigenvalue, the population may not reach a steady state, and instead, it may continue to fluctuate indefinitely.

Furthermore, the lack of a dominant eigenvalue can also affect the stability of the population. In a model with a dominant eigenvalue, the population will return to its equilibrium state after a disturbance, such as a natural disaster. However, in a model with no dominant eigenvalue, the population may not return to its previous state, and the effects of the disturbance may be amplified or dampened, leading to unpredictable population dynamics.

In conclusion, the absence of a dominant eigenvalue in a Leslie matrix can have significant consequences on a population model, such as unpredictable population fluctuations, the inability to reach a steady state, and unstable population dynamics. I hope this helps answer your question. For further reading on this topic, I suggest looking into nonlinear dynamics and chaos theory, as these concepts are closely related to the behavior of systems with no dominant eigenvalue.