Figure 2

Dynamical behavior of the system ( 2.1 ) with \(\pmb{Q=0.08}\) , \(\pmb{N^{0}=20}\) , \(\pmb{D=0.1}\) , \(\pmb{\mu=0.1}\) , \(\pmb{\delta=0.5}\) , \(\pmb{K=1.8}\) , \(\pmb{m=0.9}\) , \(\pmb{d=0.0001}\) , \(\pmb{N_{1}(0)=18}\) , \(\pmb{N_{2}(0)=0.2}\) , \(\pmb{x(0)=0.1}\) , \(\pmb{R=0.39<1}\) . (a) Time-series of the nutrient concentration outside the rhizosphere (patch 2). (b) Time-series of the nutrient concentration inside the rhizosphere (patch 1). (c) Phase portrait denoting the rhizosphere microbe-eradication periodic solution \((\bar{N}_{1}^{*}(t), \bar{N}_{2}^{*}(t),0)\) is globally asymptotically stable for \(R<1\).