Dynamic Properties of a Differential-Algebraic Biological Economic System

<table>When <svg height="11.225" id="M172" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 72.425003 11.225" width="72.425003" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.963)"><use xlink:href="#x1D70F"></use></g><g transform="matrix(.017,-0,0,-.017,12.982,10.963)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,27.685,10.963)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,35.845,10.963)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,39.721,10.963)"><path d="M153 550l-26 -186q79 31 111 31q90 0 141.5 -51t51.5 -119q0 -93 -89 -166q-85 -69 -173 -71q-32 0 -61.5 11.5t-41.5 23.5q-18 17 -17 34q2 16 22 33q14 9 26 -1q61 -50 124 -50q60 0 93 43.5t33 104.5q0 69 -41.5 110t-121.5 41q-53 0 -102 -20l38 305h286l6 -8
l-26 -65h-233z" id="x35"></path></g><g transform="matrix(.017,-0,0,-.017,47.88,10.963)"><path d="M137 343l67 33q37 17 63 17q79 0 129.5 -53t50.5 -131q0 -92 -58 -156.5t-147 -64.5t-147 68t-58 182q0 63 17 119t43 95.5t61.5 72t69 52t67.5 31.5q62 22 128 33l6 -32q-56 -11 -108 -35q-149 -71 -184 -231zM227 337q-47 0 -95 -27q-6 -23 -6 -70q0 -93 36 -155.5
t96 -62.5q53 0 78 45.5t25 105.5q0 68 -35 116t-99 48z" id="x36"></path></g><g transform="matrix(.017,-0,0,-.017,56.039,10.963)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,64.199,10.963)"><path d="M249 635q70 0 116 -43t46 -105q0 -46 -28 -80q-22 -25 -80 -64q62 -35 97 -75t35 -99q0 -81 -63 -131t-135 -50q-83 0 -137.5 45.5t-54.5 123.5q0 52 45 95q29 28 89 64q-109 62 -109 155q0 66 50.5 115t128.5 49zM238 603q-42 0 -67.5 -31t-25.5 -72q0 -50 32.5 -79.5
t98.5 -62.5q61 48 61 124q0 59 -29.5 90t-69.5 31zM248 20q46 0 76.5 33.5t30.5 89.5q0 50 -39 85.5t-110 71.5q-81 -54 -81 -137q0 -67 35.5 -105t87.5 -38z" id="x38"></path></g>
</svg>, periodic solutions occur from <svg height="15.4" id="M173" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 18 15.4" width="18" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D44C"></use></g>
<g transform="matrix(.012,-0,0,-.012,11.6,15.187)"><use xlink:href="#x30"></use></g>
</svg>.</table>

Journal of Applied Mathematics

fig2

Figure 2

Figure 2: Dynamic Properties of a Differential-Algebraic Biological Economic System 