Periodic Property and Asymptotic Behavior for a Discrete Ratio-Dependent Food-Chain System with Delays

<div>The time histories of <span class="nowrap"><svg height="10.9953pt" id="M225" style="vertical-align:-3.18148pt" version="1.1" viewbox="-0.0498162 -7.81382 30.1953 10.9953" width="30.1953pt" 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="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"></path></g><g transform="matrix(.013,0,0,-0.013,7.339,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,17.875,0)"><path d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 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Discrete Dynamics in Nature and Society

fig1

Figure 1

Figure 1: Periodic Property and Asymptotic Behavior for a Discrete Ratio-Dependent Food-Chain System with Delays 