Let ${b_{j}}_{j=1}^{k}$ be meromorphic functions, and let $w$ be admissible meromorphic solutions of delay differential equation $$w(z)=w(z)left[frac{P(z, w(z))}{Q(z,w(z))}+sum_{j=1}^{k}b_{j}(z)w(z-c_{j})right]$$ with distinct delays $c_{1}, ldots, c_{k}inmathbb{C}setminus{0},$ where the two nonzero polynomials $P(z, w(z))$ and $Q(z, w(z))$ in $w$ with meromorphic coefficients are prime each other. We obtain that if $limsup_{rrightarrowinfty}frac{log T(r, w)}{r}=0,$ then $$deg_{w}(P/Q)leq k+2.$$ Furthermore, if $Q(z, w(z))$ has at least one nonzero root, then $deg_{w}(P)=deg_{w}(Q)+1leq k+2;$ if all roots of $Q(z, w(z))$ are nonzero, then $deg_{w}(P)=deg_{w}(Q)+1leq k+1;$ if $deg_{w}(Q)=0,$ then $deg_{w}(P)leq 1.$par In particular, whenever $deg_{w}(Q)=0$ and $deg_{w}(P)leq 1$ and without the growth condition, any admissible meromorphic solution of the above delay differential equation (called Lenhart-Travis type logistic delay differential equation) with reduced form can not be an entire function $w$ satisfying $overline{N}(r, frac{1}{w})=O(N(r, frac{1}{w}));$ while if all coefficients are rational functions, then the condition $overline{N}(r, frac{1}{w})=O(N(r, frac{1}{w}))$ can be omitted. Furthermore, any admissible meromorphic solution of the logistic delay differential equation (that is, for the simplest special case where $k=1$ and $deg_{w}(P/Q)=0$ ) satisfies that $N(r,w)$ and $T(r, w)$ have the same growth category. Some examples support our results.
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