In this study, monoclonal population growth with piecewise constant arguments is modeled, where ⟦t⟧ and ⟦t-1⟧ are embedded as coefficients to equation (A) to emphasize the population growth for specific times such as\ndx(t)/dt=x(t)r(1-αx(t)-β_0 ⟦t⟧x(⟦t⟧)-β_1 ⟦t-1⟧x(⟦t-1⟧)). (A)\nThe parameters α,β_0,β_1 moreover, r belongs to R^+ and ⟦t⟧ is the integer part of t∈[0,∞). Parameter r is the population growth rate of the monoclonal tumor, while α,β_0 and β_1 are rates for the delayed tumor volume that based on the logistic population model. In order to analyze the dynamic behavior of the equation (A), we study the conditions for local and global stability. In addition, it is proven that the discrete equation undergoes period doubling (flip) and Neimark-Sacker bifurcation. For early detection of monoclonal tumor growth, we incorporate an Allee function of time t and analyze the behavior for a strong Allee effect.

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