We investigate the global stability and the convergence rate of the exponential model: un+1=λ4un−2e−vn+λ3un−1+λ2un+λ1,$$ {u}_{n#x0002B;1}#x0003D;{lambda}_4{u}_{n-2}{e}#x0005E;{-{v}_n}#x0002B;{lambda}_3{u}_{n-1}#x0002B;{lambda}_2{u}_n#x0002B;{lambda}_1, $$ vn+1=μ4vn−2e−un+μ3vn−1+μ2vn+μ1,$$ {v}_{n#x0002B;1}#x0003D;{mu}_4{v}_{n-2}{e}#x0005E;{-{u}_n}#x0002B;{mu}_3{v}_{n-1}#x0002B;{mu}_2{v}_n#x0002B;{mu}_1, $$ where n=0,1,⋯$$ n#x0003D;0,1,cdots $$. The initials u0,v−2,u−1,v−1,v0$$ {u}_0,{v}_{-2},{u}_{-1},{v}_{-1},{v}_0 $$, and u−2$$ {u}_{-2} $$ and the parameters μ1,λ1,λ2,μ3,$$ {mu}_1,{lambda}_1,{lambda}_2,{mu}_3, $$ λ3,μ2,λ4$$ {lambda}_3,{mu}_2,{lambda}_4 $$, and μ4$$ {mu}_4 $$ are non‐negative real numbers. We also discuss the unboundedness, persistence, and boundedness of this system. Moreover, we introduce conditions for uniqueness and existence of the equilibrium. Finally, we give numerical explanations to verify our results. We can use the above system as a model for the growth of some perennial plants and their relationships with each other.
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