一類具有季節(jié)交替的n維Gilpin-Ayala競(jìng)爭(zhēng)模型的動(dòng)力學(xué)
2024-06-25 00:00:00陳梅香謝溪莊
摘要：研究一類具有季節(jié)交替的n維Gilpin-Ayala競(jìng)爭(zhēng)模型。利用單調(diào)動(dòng)力系統(tǒng)的理論，當(dāng)n=1時(shí)，系統(tǒng)存在著閾值動(dòng)力學(xué)。根據(jù)離散競(jìng)爭(zhēng)映射的負(fù)載單形理論，證得n維系統(tǒng)存在一個(gè)（n-1）維的負(fù)載單形。結(jié)果表明：（n-1）維的負(fù)載單形吸引了系統(tǒng)在Rn+中的所有非平凡軌道。
關(guān)鍵詞：季節(jié)交替；Gilpin-Ayala競(jìng)爭(zhēng)模型；周期解；龐加萊映射；負(fù)載單形
中圖分類號(hào)：O 175.13文獻(xiàn)標(biāo)志碼：A
文章編號(hào)：1000-5013（2024）03-0417-06
Dynamics of A n-Dimensional Gilpin-Ayala Competition Model With Seasonal Succession
CHEN Meixiang，XIE Xizhuang
（School of Mathematical Sciences，Huaqiao University，Quanzhou 362021，China）
Abstract：A type of n dimensional Gilpin-Ayala competition models with seasonal succession are studied. Using the theory of monotonic dynamical systems，when n=1，the system has threshold dynamics. Using the theory of carrying simplex of discrete competitive mappings，the existence of a （n-1） dimensional carrying simplex in the n dimensional system is proved. The result shows that （n-1） dimensional carrying simplex attracts all nontrivial orbits in Rn+ of the system.
Keywords：seasonal succession；Gilpin-Ayala competition model；periodic solution；Poincaré mapping；carrying simplex
1 預(yù)備知識(shí)
季節(jié)性更替是自然界的普遍現(xiàn)象，深深影響著種群的生存與增長(zhǎng)，群落的結(jié)構(gòu)和組成[1]。當(dāng)氣溫、降水量、氣壓、濕度和季風(fēng)隨著季節(jié)的更替而變化時(shí)，種群和群落處于一個(gè)周期性波動(dòng)的外部環(huán)境中[2-3]。Sommer等[4]利用季節(jié)交替模型研究種群動(dòng)力學(xué)[5-7]。在經(jīng)典的n種群Gilpin-Ayala競(jìng)爭(zhēng)模型[8-9]的基礎(chǔ)上，利用文獻(xiàn)[2，5]中的建模方法，構(gòu)造具有季節(jié)交替的n種群Gilpin-Ayala競(jìng)爭(zhēng)模型，即
2 基本定義和引理
3 負(fù)載單形的存在性及其證明
4 結(jié)論
1）當(dāng)n=1時(shí)，系統(tǒng)（1）存在閾值動(dòng)力學(xué)，即當(dāng)rφ-λ（1-φ）≤0時(shí)，不管種群的初始數(shù)量處于什么水平，種群都將走向滅絕；當(dāng)rφ-λ（1-φ）gt;0時(shí)，系統(tǒng)（1）存在唯一的正周期解，使種群的初始數(shù)量為非零值時(shí)，最終都將收斂到這個(gè)正周期解。
2）當(dāng)n≥2時(shí)，系統(tǒng)（1）必將存在一個(gè)（n-1）維的有界不變閉曲面（負(fù)載單形），其吸引了系統(tǒng)（1）的所有非平凡軌道。
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（責(zé)任編輯：陳志賢 "英文審校：黃心中）
