A Three-Species Food Chain System with Two Types of  Functional Responses

<table>Period-halving bifurcation: (a) attractor with period-2 when <svg height="14.6" id="M298" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 61.275002 14.6" width="61.275002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.6)">
<g transform="translate(72,-60.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑑</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.3,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,-56.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="18.279387" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="21.405136" y="0">4</tspan>
<tspan style="font-size: 12.50px; " x="27.656635" y="0">2</tspan>
</text>
</g>
</g>
</svg>; (b) attractor with period-1 when <svg height="14.6" id="M299" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 61.275002 14.6" width="61.275002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.6)">
<g transform="translate(72,-60.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑑</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.3,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,-56.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="18.279387" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="21.405136" y="0">4</tspan>
<tspan style="font-size: 12.50px; " x="27.656635" y="0">7</tspan>
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</g>
</g>
</svg>.</table>

Abstract and Applied Analysis

fig7

Figure 7

Figure 7: A Three-Species Food Chain System with Two Types of  Functional Responses 