邏輯斯族群動態機率模式參數的估計

The estimate of parameters of logistic population dynamic probability model

陳啟天(Chi-Tian Chen)

指導教授 ： 彭雲明

國立臺灣大學/生物資源暨農學院/農藝學研究所/碩士(2006年)

https://doi.org/10.6342/NTU.2006.00652

摘要

邏輯斯生長模式分為決定性模式(deterministic model)與隨機性模式(stochastic model)，前者由Verhulst (1938)、Pearl和Reed (1920)提出，後者則由Bartlett等人(1960)提出，廣泛的應用在族群個體數的動態描述。邏輯斯生長模式中的參數，可分為族群內生的出生及死亡率 (intrinsic rates)及族群中的擁擠係數 (crowding coefficients)。Renshaw (1991)針對非重複的族群增長資料提出參數估計的程序，對於重複資料的部分卻毫無著墨，而且其參數估計的過程，似乎有些瑕疵，所得到的參數估計值也並非相當的精確。有鑑於此，文中利用反應曲面法(response surface method)，建構出平衡解變異數( )對內生出生率及擁擠係數的反應曲面，然後利用非線性混合模式(nonlinear mixed-effect model, NME model)估計環境容納量(carrying capacity)的變異數，以此估值橫切反應曲面，投射至二維的平面上，藉此方式找出所對應的生長模式參數的估計值。

資料分析針對AHB模擬資料及Gause的草履蟲培養資料，建立高描述能力的反應曲面，NME method精確的估計 ，並描繪出 的投射線段，明顯縮小參數的選擇範圍，至於如何於線段選取適當的參數估計值，有待進一步的研究。

關鍵字

邏輯斯生長模式 ； 環境容納量 ； 內生出生及死亡率 ； 擁擠係數 ； 反應曲面法 ； 非線性混合模式

並列摘要

The deterministic logistic model has long been used for modeling the growth of animal population. However, its stochastic counterpart is less known to most of the ecologists partly due to the difficulty in parameter estimation. The estimation method given by Renshaw (1991) was rough and did not take into consideration of repeated measurement data. In this research, we propose an alternative to the estimation of the four parameters of stochastic logistic model.

Our approach includes two major steps. First part of our estimation method is to construct a relationship between the variance of quasi-equilibrium distribution and four parameters which are denoted by a1, a2, b1 and b2. The a’s are intrinsic parameters for birth and death, respectively; the b’s are so-called crowding coefficients. We fit a second-order response surface function of variances σ2 on various probable values of parameters (a1, b1) and denote the function by σ2 = f (a1, b1). Secondly, we employ the method of nonlinear mixed-effects (NME) model to the real growth data usually collected in laboratory or field by scientists. The carrying capacity K of a stochastic logistic model is viewed as a normal random variable whose mean and variance can be estimated by NME method. The variance estimated by this method is far more precise than that obtained by Renshaw’s way of estimation. By plugging the variance estimate into the previously constructed response surface function, we may ‘calibrate’ the possible values of (a1, b1) in a confined interval. We can proceed further by incorporate the mean estimate = (a1－a2)/ (b1＋b2) and other source of information such as probable skewness to determine a reasonable range of parameter estimates.

Six examples are used to illustrate and justified the newly proposed approach; they are from renowned authors such as Gause (1938), Pielou (1977), Renshaw (1991) and Matis et al (1996). Interesting results reveal that parameter estimates found by previous authors are all in the confined intervals obtained from our method.

並列關鍵字

deterministic logistic model ； stochastic logistic model ； quasi-equilibrium distribution ； intrinsic parameters for birth and death ； crowding coefficients ； nonlinear mixed-effects (NME) model

參考文獻

Bartlett, M.S., Gower, J.C., Leslie, P.H., 1960. A comparison of theoretical and empirical results for some stochastic population models. Biometrika 47, 1–11.

Braun, M., 1991. Differential Equations and Their Applications. Springer, New York.

Gause, G.F., 1938. The Struggle For Existence. Dover Publications, INC. Mineola, New York.

Gillman, M., Hails, R., 1997. An Introduction to Ecological Modeling. Blackwell Science, Inc.

Matis, J.H., Kiffe, T.R., 1996. On approximating the moments of the equilibrium distribution of a stochastic logistic model. Biometrics 52, 980–991.