Abstract
We examine the mathematical theory behind a 3- stage population matrix model that is familiar to professionals in Natural Resource and Wildlife Management (Silvertown & Franco 1989, Salguero-Gómez 2011). The projection matrix corresponds to any wildlife (or plant) population containing juvenile (seed), subadult (rosette), and adult (flowering plant) stages. The 3- stage population matrix models allows for 511 (2^9-1) possible life history strategies. We develop an alternative method to approach the standard calculations of the eigenvalue, eigenvectors, damping ratio, sensitivities, and elasticities. In the characteristic equation, the nine demographic parameters (fertilities and stage transitions) collapse into at most three superparameters. The superparameters can be used as a new avenue to address numerous problems that have stymied analytical researchers for over three decades. Eigenvalues and functions of eigenvalues can be calculated using at most three superparameters only, potentially simplifying data collection efforts for estimating growth rate and demographic quantities. Analytical expressions for the sensitivities and elasticities of the subdominant eigenvalues, cross partials, and the derivative of the damping ratio can also be calculated using only superparameter inputs. The superparameters make available a host of graphical techniques that aid theorists and practitioners in understanding the function of the matrix operator. These graphical representations also aid in closing the translation gap between matrix model theorists and practitioners (Starfield 1997). The graphics motivate a paradigm shift in how theorists may think about, and how practitioners may conduct sensitivity analysis. Through superparameters, sensitivity analysis to assess any perturbation (or set of perturbations) in any 3- stage model becomes as simple as reading a three dimensional graph. The results will be useful in propelling a new wave of theoretical research regarding transient and asymptotic time dynamics. As well, the results are useful in applied field applications, such as population viability analysis, population recovery planning, translocation planning, hunting and harvest management, policy designations and assessments.