Optimal Harvesting Policies for a Stochastic Food-Chain System with Markovian Switching

<div>Sample path of <svg height="12.2089pt" id="M247" style="vertical-align:-2.503pt" version="1.1" viewbox="-0.0498162 -9.7059 19.4248 12.2089" width="19.4248pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M425 682C425 693 422 699 407 699C369 699 308 677 260 646C213 651 191 674 191 701C191 715 195 726 200 734L183 740C153 730 118 709 118 681C118 644 163 619 218 615C168 582 115 534 115 463C115 418 147 382 180 369C102 329 23 261 23 163C23 74 74 38 158 14C183 7 201 2 221 -2C279 -15 290 -33 290 -58C290 -94 251 -145 230 -169L252 -189C280 -171 367 -103 367 -25C367 20 333 51 266 68C245 73 218 80 191 89C139 106 107 135 107 192C107 261 161 329 221 355C248 350 281 347 314 347C330 347 370 388 370 406C370 418 366 422 352 422C321 422 280 413 255 407C218 413 192 448 192 493C192 544 229 588 266 610C309 610 354 615 376 623C397 631 425 651 425 682Z" id="g113-236"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="448" vert-adv-y="448"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,5.825,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,10.323,0)"><path d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z" id="g113-117"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="341" vert-adv-y="341"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,14.756,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g></svg> with initial condition <svg height="12.2089pt" id="M248" style="vertical-align:-2.503pt" version="1.1" viewbox="-0.0498162 -9.7059 42.4253 12.2089" width="42.4253pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M425 682C425 693 422 699 407 699C369 699 308 677 260 646C213 651 191 674 191 701C191 715 195 726 200 734L183 740C153 730 118 709 118 681C118 644 163 619 218 615C168 582 115 534 115 463C115 418 147 382 180 369C102 329 23 261 23 163C23 74 74 38 158 14C183 7 201 2 221 -2C279 -15 290 -33 290 -58C290 -94 251 -145 230 -169L252 -189C280 -171 367 -103 367 -25C367 20 333 51 266 68C245 73 218 80 191 89C139 106 107 135 107 192C107 261 161 329 221 355C248 350 281 347 314 347C330 347 370 388 370 406C370 418 366 422 352 422C321 422 280 413 255 407C218 413 192 448 192 493C192 544 229 588 266 610C309 610 354 615 376 623C397 631 425 651 425 682Z" id="g113-236"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="448" vert-adv-y="448"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,5.825,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,10.323,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g121-46"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,16.563,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,24.693,0)"><path d="M535 397V447H52V397H535ZM535 230V280H52V230H535ZM535 63V113H52V63H535Z" id="g117-35"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,35.956,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g121-47"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg>.</div>

Mathematical Problems in Engineering

fig3

Figure 3

Figure 3: Optimal Harvesting Policies for a Stochastic Food-Chain System with Markovian Switching 