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A set of sufficient conditions is obtained for the global attractivity of the following two-species discrete mutualism model with infinite deviating arguments: x1(k+1)=x1kexp⁡r1(K1+α1∑s=0+∞J2sx2k-s)/(1+∑s=0+∞J2sx2k-s)-x1k and x2(k+1)=x2kexp⁡r2(K2+α2∑s=0+∞J1sx1k-s)/(1+∑s=0+∞J1sx1k-s)-x2k, where ri,Ki,αi, i=1,2, are all positive constants, ∑j=1+∞Ji(n)=1, and αi>Ki. Our results generalize the main result of Yang et al. (2014).

We discuss a discrete mutualism model with variable delays of the formsN1(n+1)=N1(n)exp{r1(n)[(K1(n)+α1(n)N2(n-μ2(n)))/1+N2(n-μ2(n)))-N1(n-ν1(n))]},N2(n+1)=N2(n)exp{r2(n)[(K2(n)+α2(n)N1(n-μ1(n)))/(1+N1(n-μ1(n)))-N2(n-ν2(n))]}. By means of an almost periodic functional hull theory, sufficient conditions are established for the existence and uniqueness of globally attractive almost periodic solution to the previous system. Our results complement and extend some scientific work in recent years. Finally, some examples and numerical simulations are given to illustrate the effectiveness of our main results.

We propose and deal with a discrete mutualism model with infinite deviating arguments and feedback controls. Sufficient conditions which guarantee the permanence of the system are obtained by using the difference inequality theory. The paper ends with brief conclusions.

We propose a discrete model of mutualism with infinite deviating arguments, that is 𝑥 1 ( 𝑛 + 1 ) = 𝑥 1 ( 𝑛 ) e x p { 𝑟 1 ( 𝑛 ) [ ( 𝐾 1 ( 𝑛 ) + 𝛼 1 ∑ ( 𝑛 ) ∞ 𝑠 = 0 𝐽 2 ( 𝑠 ) 𝑥 2 ∑ ( 𝑛 − 𝑠 ) ) / ( 1 + ∞ 𝑠 = 0 𝐽 2 ( 𝑠 ) 𝑥 2 ( 𝑛 − 𝑠 ) ) − 𝑥 1 ( 𝑛 − 𝜎 1 ( 𝑛 ) ) ] } , 𝑥 2 𝑥 ( 𝑛 + 1 ) = 2 ( 𝑛 ) e x p { 𝑟 2 ( 𝑛 ) [ ( 𝐾 2 ( 𝑛 ) + 𝛼 2 ∑ ( 𝑛 ) ∞ 𝑠 = 0 𝐽 1 ( 𝑠 ) 𝑥 1 ∑ ( 𝑛 − 𝑠 ) ) / ( 1 + ∞ 𝑠 = 0 𝐽 1 ( 𝑠 ) 𝑥 1 ( 𝑛 − 𝑠 ) ) − 𝑥 2 ( 𝑛 − 𝜎 2 ( 𝑛 ) ) ] } . By some Lemmas, sufficient conditions are obtained for the permanence of the system.