# Calculate the period of prey and predator populations using the approximation if T = 7.25

Calculate the period of prey and predator populations using the approximation if $T=7.25$
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Step 1
The given equation of predator population
$y=\frac{aK}{\alpha }\sqrt{\frac{c}{a}}\mathrm{sin}\left(\sqrt{act}\right)$. Where y is the population of the prey and predator population with negligible initial condition.
The given equation of predator population is $x=\frac{cK}{\gamma }\mathrm{cos}\left(\sqrt{act}\right).$
Where x is the population of the prey with negligible initial condition.
From the sinusoidal function $y={y}_{m}\mathrm{sin}\left(\omega t\right)$ the time period can be calculated as $T=2\pi .$
By comparing the given equation with sinusoidal function, the time period is $T=\frac{2\pi }{\sqrt{ac}}$
Taking into consideration the values of the system, $c=0.75,a=1,\alpha =0.5,\gamma =0.25.$
Then calculation of time period is $T=\frac{2\pi }{\sqrt{ac}}$
$T=\frac{2\pi }{\sqrt{ac}}$
$=\frac{2\pi }{\sqrt{1×0.75}}$
$\approx 7.25$
Therefore, the period of the oscillations of the prey and predator population is $T=7.25.$