"A population that grows logistically, initially increases exponentially; then the growth lows down and eventually approaches an upper bound or limit. The most well-known form of the model is the logistic differential equation." (Linda J S Allen, "An Introduction to Mathematical Biology", 2007)
"An important question that we will address with respect to biological models is, 'what is the asymptotic or long-term behavior of the model'? For models formulated in terms of linear difference equations, the asymptotic behavior depends on the eigenvalues, whether the eigenvalues are real or complex and the magnitude of the eigenvalues. To address this question, it is generally not necessary to find explicit solutions. In cases where there exists an eigenvalue whose magnitude exceeds all others, referred to as a strictly dominant eigenvalue, then this eigenvalue is an important determinant of the dynamics." (Linda J S Allen, "An Introduction to Mathematical Biology", 2007)
"Partial differential equations arise in biological systems because the quantity being modeled not only changes continuously with respect to time but changes continuously with respect to another variable such as age or spatial location." (Linda J S Allen, "An Introduction to Mathematical Biology", 2007)
"Global stability of an equilibrium removes the restrictions on the initial conditions. In global asymptotic stability, solutions approach the equilibrium for all initial conditions. [...] In a study of local stability, first equilibrium solutions are identified, then linearization techniques are applied to determine the behavior of solutions near the equilibrium. If the equilibrium is stable for any set of initial conditions, then this type of stability is referred to as global stability." (Linda J S Allen, "An Introduction to Mathematical Biology", 2007)
"It is important to distinguish between local and global stability. Local stability of an equilibrium implies that solutions approach the equilibrium only if they arc initially close to it. For example, if the initial population size is very small and the zero equilibrium is stable, then extinction of the population may occur. however, it the initial population size is large, then local stability of the zero equilibrium tells nothing about population extinction. Global stability of an equilibrium is much stronger. Global stability implies that regardless of the initial population size, solutions approach the equilibrium. We state conditions for local stability and global stability of an equilibrium in the case or a scalar difference equation, where only one state is modeled such as population size. In addition, we state conditions for local stability of an equilibrium when several states are modeled by first-order difference equations or when one state is modeled by a second-order or higher-order difference equation. These latter conditions are known as the Jury conditions." (Linda J S Allen, "An Introduction to Mathematical Biology", 2007)
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