Let τ ∈ ℂ {0}; let p and q be distinct positive integers, and let a; b; c be meromorphic functions such that at least one of b and c is not identically equal to zero. The main purpose of this paper is to study the logistic delay differential equations of the Lotka-Volterra type <Equation ID="Equ1"> <EquationSource Format="TEX">$w^{\prime}(z)=w(z)[a(z)+b(z)w^{p}(z-\tau)+c(z)w^{q}(z-\tau)].$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mi>w</mi> <mrow> <mi mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>w</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mi>w</mi> <mrow> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mi>w</mi> <mrow> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </math></EquationSource> </Equation>We prove that any admissible meromorphic solution w of the equation satisfies that the counting function N(r; w) of poles and the characteristic function T(r; w) have the same growth category. Furthermore, we obtain that “most” of admissible meromorphic solutions of a more general delay differential equation <Equation ID="Equ2"> <EquationSource Format="TEX">$w^{\prime}(z)=w(z)\left[a(z)+\sum_{j=1}^{k}b_{j}(z)w^{j}(z-\tau)\right],\quad k\in \mathbb{N},$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mi>w</mi> <mrow> <mi mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>w</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> </mrow> </munderover> <msub> <mi>b</mi> <mrow> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mi>w</mi> <mrow> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>]</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> </math></EquationSource> </Equation> have a pole at least.

Meromorphic Solutions of Logistic Delay Differential Equations of the Lotka-Volterra Type and Beyond

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发表日期：

2026年1月2日

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Ling Xu,
Run-zi Luo,
Ting-bin Cao

摘要

Let τ ∈ ℂ {0}; let p and q be distinct positive integers, and let a; b; c be meromorphic functions such that at least one of b and c is not identically equal to zero. The main purpose of this paper is to study the logistic delay differential equations of the Lotka-Volterra type $w^{\prime}(z)=w(z)[a(z)+b(z)w^{p}(z-\tau)+c(z)w^{q}(z-\tau)].$ $w^{'} (z) = w (z) [a (z) + b (z) w^{p} (z - τ) + c (z) w^{q} (z - τ)] .$

We prove that any admissible meromorphic solution w of the equation satisfies that the counting function N(r; w) of poles and the characteristic function T(r; w) have the same growth category. Furthermore, we obtain that “most” of admissible meromorphic solutions of a more general delay differential equation $w^{\prime}(z)=w(z)\left[a(z)+\sum_{j=1}^{k}b_{j}(z)w^{j}(z-\tau)\right],\quad k\in \mathbb{N},$ $w^{'} (z) = w (z) [a (z) + \sum_{j = 1}^{k} b_{j} (z) w^{j} (z - τ)], k \in N,$ have a pole at least.