Global Dynamics of an Exploited Prey-Predator Model with Constant Prey Refuge

<table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td align="left"><span style="width: 13.5375px;"><svg height="7.9499998" id="M215" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 13.5375 7.9499998" width="13.5375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6
q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z" id="x1D45A"></path></g>
</svg></span></td><td align="center"><span style="width: 17.3125px;"><svg height="17.9125" id="M216" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 17.3125 17.9125" width="17.3125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,17.65)"><path d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37
q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" id="x1D465"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.563,9.488)"><path d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z" id="x2032"></path></g><g transform="matrix(.012,-0,0,-.012,13.096,9.488)"><path d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z" id="x2032"></path></g>
</svg></span></td><td align="center"><span style="width: 17.6625px;"><svg height="22.15" id="M217" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 17.6625 22.15" width="17.6625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,17.65)"><path d="M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26
q13 28 43 28q18 0 30 -15.5t12 -39.5z" id="x1D466"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.9,9.488)"><path d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z" id="x2032"></path></g><g transform="matrix(.012,-0,0,-.012,13.434,9.488)"><path d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z" id="x2032"></path></g>
</svg></span></td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td align="left">0.1</td><td align="center">1.03578</td><td align="center">3.0847</td></tr><tr><td align="left">0.2</td><td align="center">1.13578</td><td align="center">3.37539</td></tr><tr><td align="left">0.398969</td><td align="center">1.33578</td><td align="center">3.95004</td></tr><tr><td align="left">0.6</td><td align="center">1.53578</td><td align="center">4.5256</td></tr><tr><td align="left">0.75</td><td align="center">1.68578</td><td align="center">4.95175</td></tr><tr><td align="left">1</td><td align="center">1.93578</td><td align="center">5.65572</td></tr><tr><td align="left">3</td><td align="center">3.93578</td><td align="center">11.0052</td></tr><tr><td align="left">10</td><td align="center">10.93578</td><td align="center">25.7753</td></tr><tr><td align="left">20</td><td align="center">20.93578</td><td align="center">36.2088</td></tr><tr><td align="left">23.31422*</td><td align="center">24.25002</td><td align="center">36.8980</td></tr><tr><td align="left">24</td><td align="center">24.93578</td><td align="center">36.8685</td></tr><tr><td align="left">25</td><td align="center">25.93578</td><td align="center">36.7197</td></tr><tr><td align="left">30</td><td align="center">30.93578</td><td align="center">34.0934</td></tr><tr><td align="left">35</td><td align="center">35.93578</td><td align="center">28.3297</td></tr><tr><td align="left">40</td><td align="center">40.93578</td><td align="center">19.4288</td></tr><tr><td align="left">47</td><td align="center">47.93578</td><td align="center">7.39071</td></tr><tr><td align="left">47.5</td><td align="center">48.43578</td><td align="center">0.195172</td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr><tr class="table-fn"><td><svg height="13.75" id="M218" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 7.7375002 13.75" width="7.7375002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.012,-0,0,-.012,.062,5.525)"><path d="M471 153q-22 -15 -61 -13q-25 31 -45.5 49.5t-56.5 41.5q4 -71 28 -134q-17 -33 -42 -46q-24 12 -42 46q24 63 28 134q-36 -23 -56.5 -41.5t-45.5 -49.5q-36 -2 -61 13q0 28 19 59q65 10 130 43q-65 33 -130 43q-19 31 -19 59q22 15 61 13q25 -31 45.5 -49.5t56.5 -41.5
q-4 71 -28 134q17 33 42 46q24 -12 42 -46q-24 -63 -28 -134q36 23 56.5 41.5t45.5 49.5q36 2 61 -13q0 -28 -19 -59q-65 -10 -130 -43q65 -33 130 -43q19 -31 19 -59z" id="x2217"></path></g>
</svg>At <svg height="7.9499998" id="M219" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 13.5375 7.9499998" width="13.5375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6
q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z" id="x1D45A"></path></g>
</svg> = 23.31422, the value of <svg height="22.15" id="M220" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 17.6625 22.15" width="17.6625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,17.65)"><path d="M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26
q13 28 43 28q18 0 30 -15.5t12 -39.5z" id="x1D466"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.9,9.488)"><path d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z" id="x2032"></path></g><g transform="matrix(.012,-0,0,-.012,13.434,9.488)"><path d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z" id="x2032"></path></g>
</svg> is maximum.<br/></td></tr></table>

International Scholarly Research Notices

tab1

Table 1

Table 1: Global Dynamics of an Exploited Prey-Predator Model with Constant Prey Refuge 