مدل رشد اقتصادی با ظرفیت حمل درونزا و گذار جمعیت شناسی

In this paper, a mathematical model is set up to inquire population change under interaction between the economic growth and human population carrying capacity. By introducing the population growth equation with variable carrying capacity into the classical Solow model and combining the population growth equation, we obtain a two-dimensional dynamical system. It is proved that the dynamical system has a unique equilibrium and the solution of the dynamical system is asymptotically stable. By qualitative analysis, we obtain that the population growth rate increases from zero to a positive level firstly and then decreases to zero and per capita capital increases strictly along a normal economic growth path. Therefore, the model implies that the demographic transition appears under the interaction between economic growth and human population carrying capacity.

The affection of the demographic transition and the carrying capacity (CC) of human population on economic growth has aroused much attention by demographers, economists and biologists, recently [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] and [11]. In general, the population carrying capacity is assumed to be fixed or change with the time and the growth of population satisfies logistic equation or the equation of logistically variable carrying capacity [12], [13] and [14]. However, the population carrying capacity expands with the economic growth and the population, and the labor increases, which accelerates the economic growth, as the population carrying capacity expands. So, there exists interaction between the economic growth and the population carrying capacity. In this paper, we assume that the population carrying capacity increases with the economic growth, and the population grows as the population carrying capacity expands. By integrating the variable population carrying capacity function into the classical Solow model and combining the population growing equation, we obtain a two-dimensional dynamical system. It is proved that the dynamical system has a unique nonzero equilibrium and its solution is asymptotically stable and converges to the equilibrium. By qualitative analysis, we obtain that the population growth rate increases from zero to a positive level firstly and then decreases to zero, and per capita capital increases strictly along the economic growth path that starting from a point on the curve of the variable population carrying capacity function. Therefore, the model implies that the demographic transition appears under the interaction between economic growth and human population carrying capacity. In the end of this paper, we provide a numerical simulation to show the process of the economic growth and the demographic transition. The remainder of the paper is organized as follows. In Section 2, the model is presented. The existence and uniqueness of nonzero equilibrium of the model is proved and the type of the equilibrium is discussed in Section 3 and in Section 4. In Sections 5, 6 and 7, we analyze the dynamics of the model, per capita capital and the population along the economic growth path. The numerical simulation and some conclusions are presented in Sections 8 and 9.

From Theorem 8 and Fig. 4(d), we see that the population growth rate increases firstly and then turns to decline along a normal growth path, that is, the demographic transition appears in our model along a normal growth path. In fact, there exist two stages of the population growth rate change. One stage is the period of the population growth rate increasing and the other is the period of the population growth rate declining. At the beginning of the economic growth, the population carrying capacity grows faster than the population grows and the growth rate of the population increases. As the carrying capacity tends to the equilibrium level, the growth rate of the population declines and tends to zero. By the analysis in Section 7, we see that the per capita capital (so as the per capita consumption) increases along a normal growth path in spite of the population being grown in the first stage. We also see that the dynamics of the aggregate capital and per capita capital is similar to the classical Solow model along a normal growth path except the dynamics of the population dynamics. Comparing the normal growth paths AA and BB in Section 8, we find that the population grows fast when the aggregate capital and the population carrying capacity increase fast. In fact, point AA corresponds to the high growth rate of the aggregate capital and the population carrying capacity since it starts at a low level of the initial capital and population and the function of human population carrying capacity is assumed to be concave.