ODE-06

Lotka prey slope

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ODELotka

Governing equation

x'=x(\\alpha-\\beta y)

where

\alpha: α (1/yr)
\beta: β (1/yr)
x: x (—)
y: y (—)
x': x' (1/yr)

Lecture brief

Historical brief

Exponential decay, the harmonic oscillator, SIR epidemics, Lotka–Volterra and Bessel are the first solvable ODEs of nature and populations. The lab is rate, period and phase portrait in closed form. This sheet (ODE-06 — Lotka prey slope) is the form associated with Lotka. Working symbols: $\alpha$ , $\beta$ , $x$ , $y$ $\rightarrow$ $x'$ . Mass-action hunt.

Purpose

Purpose: compute

x'

from

\alpha

\beta

x

y

in Differential equations via

x'=x(\\alpha-\\beta y)

Mass-action hunt. Use it when a real differential equations question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\alpha = 1.000\,\mathrm{1/yr}

\beta = 0.100\,\mathrm{1/yr}

x = 20.000\,\mathrm{—}

y = 8.000\,\mathrm{—}

, the governing relation

x'=x(\\alpha-\\beta y)

yields

x' = 4.000\,\mathrm{1/yr}

. Mass-action hunt. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

α \alpha1 1/yrβ \beta0.1 1/yrx x20 —y y8 —

Outputs

x' x'4.000 1/yr

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