[Paper Review] Applications of Perron-Frobenius Theory to Population Dynamics

Chi-Kwong Li, Hans Schneider|ArXiv.org|Sep 2, 2001

Mathematical and Theoretical Epidemiology and Ecology Models12 references4 citations

TL;DR

This paper applies Perron-Frobenius theory to matrix models in population dynamics, providing concise proofs of the Fundamental Theorem of Demography and refining Cushing and Yicang's theorem on the net reproductive rate $ R_0 $. It establishes that scaling the fertility matrix by $ q(s) $ achieves any desired growth rate $ s $, with $ R_0 = 1 $ corresponding to population stability when $ s = 1 $, and demonstrates how $ R_0 $ governs long-term population behavior via spectral properties of the next-generation matrix.

ABSTRACT

By the use of Perron-Frobenius theory, simple proofs are given of the Fundamental Theorem of Demography and of a theorem of Cushing and Yicang on the net reproductive rate occurring in matrix models of population dynamics. The latter result is further refined with some additional nonnegative matrix theory. When the fertility matrix is scaled by the net reproductive rate, the growth rate of the model is 1. More generally, we show how to achieve a given growth rate for the model by scaling the fertility matrix. Demographic interpretations of the results are given.

Motivation & Objective

• To provide simple, rigorous proofs of key results in population dynamics using Perron-Frobenius theory.
• To refine and generalize Cushing and Yicang’s theorem on the net reproductive rate $ R_0 $ using nonnegative matrix theory.
• To establish a functional relationship $ q(s) $ that scales the fertility matrix to achieve any specified growth rate $ s $.
• To clarify demographic interpretations of $ R_0 $, $ q(s) $, and spectral properties of the projection matrix $ P $.
• To demonstrate that $ R_0 = 1 $ corresponds to zero growth rate and population stationarity under appropriate conditions.

Proposed method

• Utilizes the Perron-Frobenius theorem for irreducible nonnegative matrices to analyze spectral properties of the projection matrix $ P = T + F $.
• Defines the next-generation matrix $ Q = F(I - T)^{-1} $, whose spectral radius $ \rho(Q) = R_0 $ is the net reproductive rate.
• Derives $ q(s) $ as a function such that scaling $ F $ by $ q(s) $ yields a model with growth rate $ s $, using $ q(s) = \text{leading entry of } F(I - T/s)^{-1}/s $.
• Applies graph-theoretic extensions of Perron-Frobenius theory to refine results for irreducible matrices.
• Uses the resolvent identity $ (I - T)^{-1} = \sum_{k=0}^\infty T^k $ under the condition $ \rho(T) < 1 $ to express $ Q $ as an infinite series of survival and reproduction events.
• Analyzes the eigenvalue equation $ q(r) = 1 $, where $ r $ is the growth rate, showing that $ r $ is the positive root of a polynomial in $ s^{-1} $.

Experimental results

Research questions

• RQ1 How can Perron-Frobenius theory be used to prove the Fundamental Theorem of Demography for primitive projection matrices?
• RQ2 What is the precise role of the net reproductive rate $ R_0 $ in determining long-term population growth or stability?
• RQ3 How can the fertility matrix be scaled to achieve a desired population growth rate $ s $, and what is the functional form of the required scaling factor $ q(s) $?
• RQ4 What is the demographic interpretation of $ R_0 = 1 $, and how does it relate to population stationarity?
• RQ5 How do spectral properties of $ Q $ and $ P $ relate when $ P $ is irreducible but imprimitive, and what implications does this have for generational stability?

Key findings

• The net reproductive rate $ R_0 = \rho(Q) $ is the spectral radius of the next-generation matrix $ Q = F(I - T)^{-1} $, representing the expected number of offspring per newborn over its lifetime.
• When $ R_0 = 1 $, the population growth rate $ r = 1 $, indicating population stationarity under a primitive projection matrix.
• For any $ s > 0 $, the fertility matrix $ F $ can be scaled by $ q(s) $ such that the resulting model has growth rate $ s $, with $ q(s) $ derived as the leading entry of $ F(I - T/s)^{-1}/s $.
• In the Leslie matrix case, $ q(s) = f_1 s^{-1} + f_2 t_1 s^{-2} + \cdots + f_n (t_{n-1} \cdots t_1) s^{-n} $, a polynomial in $ s^{-1} $, and the growth rate $ r $ satisfies $ q(r) = 1 $.
• For the example with $ T $ and $ F $ scaled such that $ \rho(T) = 0 $, the scaling function is $ q(s) = (1 + 2s^2)/(8s^4) $, and the corresponding $ R_0(s) = 3s^4 / (1 + 2s^2) $.
• Even when $ \rho(P) = 1 $, the normalized population may oscillate indefinitely, showing that generational stability does not imply convergence of population distribution.