key: cord-0486558-6cotmyru
authors: Karim, Md Aktar Ul; Bhagat, Supriya Ramdas; Bhowmick, Amiya Ranjan
title: A New Method to Determine the Presence of Continuous Variation in Parameters of Biological Growth Curve Models
date: 2021-02-16
journal: nan
DOI: nan
sha: f5c0cf0060711328d44745fb70e0520c71622156
doc_id: 486558
cord_uid: 6cotmyru

Quantitative assessment of the growth of biological organisms has produced many mathematical equations. Many efforts have been given on statistical identification of the correct growth model from experimental data. Every growth equation is unique in terms of mathematical structures; however, one model may serve as a close approximation of the other by appropriate choice of the parameter(s). It is still a challenging problem to select the best estimating model from a set of model equations whose shapes are similar in nature. Our aim in this manuscript is to develop methodology that will reduce the efforts in model selection. This is achieved by utilizing an existing model selection criterion in an innovative way that reduces the number of model fitting exercises substantially. In this manuscript, we have shown that one model can be obtained from the other by choosing a suitable continuous transformation of the parameters. This idea builds an interconnection between many equations which are scattered in the literature. We also get several new growth equations; out of them large number of equations can be obtained from a few key models. Given a set of training data points and the key models, we utilize the idea of interval specific rate parameter (ISRP) proposed by Bhowmick et al (2014) to obtain a suitable mathematical model for the data. The ISRP profile of the parameters of simpler models indicates the nature of variation in parameters with time, thus, enable the experimenter to extrapolate the inference to more complex models. Our proposed methodology significantly reduces the efforts involved in model fitting exercises. The proposed idea is verified by using simulated and real data sets. In addition, theoretical justifications have been provided by investigating the statistical properties of the estimators.

assumed the confined exponential function for K(t) to study the growth of microbial biomass under occlusion Table 1 . Here taking four growth models (a) exponential model, (b) logistic model, (c) confined exponential model and (d) theta-logistic model as the parent models and varying the parameters continuously in various ways we demonstrate the road-map. In this manner, we are also able to find some well known distribution functions, which are represented by red lettering in the above flowchart. 

Normal Distribution (Banks, 1994) 2. Power Law Exponential (Banks, 1994) 3. Gompertz Model (Banks, 1994) 6. (Korf, 1939) 7.

∞ Linearly increasing r 8.

Hyperbolically varying r 9.

Hyperbolically varying r 

Hyperbolically Varying K (Banks, 1994) 3. Chakraborty et al., 2017) 6.

We did not consider the variation r = r 0 1−ct , c > 0 (used in Banks (1994)) as it does not give biologically realistic model. Chakraborty et al., 2017) 11. r and Spina, 2016) 12. r

Second-order Exponential Polynomial(Chakraborty et al., 2019) 

2. Weibull et al., 1951) 3.

Linearly Increasing r 6.

Exponentially Decaying r 7.

K Periodically Varying r 9.

Exponentially Decaying K We did not consider the variation r = r 0 1−ct , c > 0 (used in Banks (1994)) as it does not give biologically realistic model. Logistic et al., 2015) 3. and Bajzer, 1993) 4. Theta-logistic r

nections with normal distribution, power law exponential model, linear model, hyperbolic model and gompertz model. In this paper, the extended models which are obtained by continuous transformation of the growth 162 coefficient (r) is depicted in Table 2 . In Table 3 , we considered continuously variation in intrinsic growth rate r 163 and carrying capacity K in logistic growth model. Considering continuous variation in K, Banks (1994) built 164 some connections between logistic model with other existing growth models. We mainly focus on the variation 165 in the parameter r and built connection with extended logistic model studied by Chakraborty et al. (2017) . The 166 exercises have also been carried out on the theta-logistic and confined exponential model and the derived con-167 nections are depicted in Table 4 and tained. It is to be noted that for each of the previous cases, the final differential equation (after replacing r 173 by r(t) or K by K(t)) can be solved analytically or solutions are available using some special functions. The 174 differential equations must be solved numerically to obtain the size profile in case the analytical solution is not 175 available. In Table 6 , we consider few cases in which the analytical expression for the size variable X(t) is not 176 available. The detail discussion for each extended growth equation is discussed in the supporting information. performed better in selecting the true model more accurately than the Fisher's RGR (Pal et al., 2018) . For In this sub section we shall derive the distribution of ISRP estimator of r(t) and K(t) under some assump-188 tions using Multivariate Delta Method (Wasserman, 2004; Casella and Berger, 2002) . This distribution will be 189 the key mechanism to identify whether the parameter has undergone any continuous variation over time. To derive this distribution, we consider the data matrix X, where

, whose rows are assumed to be independent and identically distributed (iid) random variables following q-193 variate normal distribution with mean vector µ = (µ t 1 , µ t 2 , . . . , µ tq ) and variance covariance matrix Σ q×q = 194 σ 2 ρ |i−j| q i,j=1 . If we take X i = (X it 1 , X it 2 , . . . , X itq ) , then E(X i ) = µ and Var(X i ) = Σ, for all i = 1, 2, . . . , n;

A denotes the transpose of a matrix A. We have considered 1, 2, . . . , q time points. We consider logistic growth 196 model as test bed model for simulation study and under this model

As existence of second order moments is assumed, so taking X = X 1 , 198 we can apply Multivariate central limit theorem (Timm, 2002) by which we get

Now the estimator of ISRP of r(t) based on time points {t j , t j+1 , t j+2 }, j = 1, 2, . . . , q −2, denoted as r j (∆t), 200 is given by (Pal et al., 2018) ,

We aim to obtain the sampling distribution of φ X j , X j+1 , X j+2 using Multivariate delta method (Wasserman, 202 2004). In terms of variables x, y and z the function φ can be written as

By using Multivariate central limit theorem for X j , X j+1 , X j+2 , we obtain

Here the mean function µ t under logistic growth law is given by eqn. (1) and φ : R 3 → R is a differentiable 205 function at µ = (µ j , µ j+1 , µ j+2 ) ∈ R 3 . Now we have to find the ∇φ at the point µ to apply Multivariate 206 delta method. Taking partial derivatives of φ(x, y, z) with respect to x, y and z and evaluate the above partial 207 differential equations at the point µ, we obtain

where the values of µ i 's, i = j, j + 1, j + 2 are provided in eqn. (1). Using matrix notation, we obtain

Now by using Multivariate delta method, the distribution of r j (∆t) is given as

The estimator of ISRP of the parameter K based on the triplet {t j , t j+1 , t j+2 }, j = 1, 2, . . . , q − 2, denoted 211 by K j (∆t), is given by (Pal et al., 2018) ,

Since K j (∆t) is a function of the random variable r j (∆t), one may argue that the distribution can be derived 213 by using the delta method with taking some function of the form g X j , X j+1 , r j (∆t) . However, note that the 214 co-variance structure between r j (∆t) and (X j , X j+1 ) is not known. Hence, the function ψ was chosen using 215 the original random variables whose covariance structure is known. In terms of (X j , X j+1 , X j+2 ), the function 216 ψ is given as

Taking partial derivatives of ψ(x, y, z) with respect to the real variables x, y and z and evaluating it at the point µ, we obtain

where the values of µ i 's, i = j, j + 1, j + 2 are provided in eqn. (1). Using matrix notation, we get

and η| µ = 1 µ 0 − ζ . By using Multivariate 221 delta method, the distribution of K j (∆t) is given as

The expression for partial derivatives which are required for the computations of interval specific estimators for 223 all the models are given in Appendix Appendix A. In this section the theoretical results are verified by using the simulation study. To check the accuracy of 226 delta method is approximating the sampling distribution of r j (∆t) for j = 1, 2, . . . , q − 2, we used computer 227 simulation. We simulated the growth trajectories for n = 1000 individuals for 20 times points where each 228 trajectory was generated from the multivariate normal distribution with logistic mean function and variance-229 covariance matrix with Koopman structure (Koopmans, 1942) . Based on this data, we obtained the estimate of 230 r j (∆t) which acts as a single realization from the sampling distribution of r j (∆t). The process was replicated 231 1000 times to obtain 1000 realizations from the distribution of r j (∆t) to visualize the distribution by using 232 the histograms. The histograms obtained from simulation study clearly suggested the agreement with normal 233 distribution with estimated mean and and variance obtained from the delta method. The close agreement 234 between the approximate distribution by using delta method and simulated sampling distribution is depicted 235 in Fig. 2a , 2b for r j (∆t) and in Fig. 2c , 2d for K j (∆t). The parameter choices are kept as K = 100, µ 0 = 236 10, σ 2 = 0.001, ρ = 0.1, and the covariance matrix is the Koopman structure.

In addition, even if the data were simulated from the other growth model, the normal approximation of the 238 sampling distribution of the estimators remain unchanged. For example, we carried out the same simulation 239 study using the extended version of the logistic growth model with r(t) = r 0 t c−1 and r(t) = r 0 (1 + ct) and Figure 3 : The figure shows the boxplot of rj(∆t) obtained from 1000 simulated realizations by using logistic growth model with continuously varying parameter r. It is to be noted that the assumed variation in r is reflected in the ISRP profile. For both the cases the simulation study was carried out using the following parameter set up as: r0 = 0.3, K = 100, µ0 = 10, ρ = 0.1 and σ 2 = 0.001. The reason for depicting only first few time points are mentioned in the main text (Discussion).

In this section we establish the utility of the discussed method using some real data sets from different 

In the third case study, we have taken the data of cumulative COVID-19 cases in Germany. The data set 294 being considered from (https://ourworldindata.org/coronavirus). The data contains cumulative affected cases 295 of COVID-19 from 31st December 2019 to 8th July 2020. Till 27th Jan 2020, there were no cases reported in 296 Germany. For the ease in the selection criterion, we have taken 5 days moving average of the data. It has been 297 observed that till day 62, dated 29th Feb 2020, there were minor changes observed in the data, due to which 298 for fitting the model the data is being considered from 1st march 2020 to 8th June 2020, which is 129 days in 299 total. For simplicity, the measurement schedules are rescaled as t = 0, 1, 2, . . . , 128.

From the size profile (Fig. 7a ) of the data, it can be clearly seen that logistic growth model seems to be 301 the best choice to start our analysis with. But from the size profile, it is hard to conclude whether or not any 302 variation is present in r (intrinsic growth rate). So for getting insight about any variation present in parameter 303 r, we plotted ISRP profile (Fig. 7b) The continuous variations of parameter r are fitted onto ISRP of r for the data by using nls2 function in software R. Since linearly growing and decaying behaves similarly for this data sets as c = 0 in both cases so, we have considered lineally growing or decaying as linearly varying. For similar reason, we have considered exponentially varying instead of exponentially growing or decaying. Note that extended logistic and linearly varying parameter gives the same fit. Panel (b): The corresponding variations of parameter r are considered in logistic growth model as in Table 3 . The analytical solutions in Table 3 are fitted onto the data to obtain the best fitted model for the data.

We have obtained the empirical estimate of ISRP of r by assuming that the data generating process was 305 logistic. To check whether there is any significant variation in r, we fitted the continuous functions given in 306   Table 3 and selected the best model based on AIC and RMSE values ( Table 7) . As there is no sign of presence 307 of periodic variation in data, we have not considered any kind of periodic variation in r as well as as in final 308 fitting. Based on the summary in Table 7 , we observed that the parameter r was subjected to hyperbolic 309 variation with respect to time (Fig. 8a) . Hence, we suspect that the actual data generation process is not Table 3 ) to the cumulative number of cases and found that the logistic model with hyperbolically varying growth coefficient is the best choice amongst all the models (both RMSE and AIC (Table 8 ) support the 313 conclusion ) (Fig. 8b) . The analysis was carried out using nonlinear least squares method using nls2 function 314 available in R. Complete source codes are provided in the supporting online material.

Based on these three case studies, we conclude that ISRP profile acts as a key indicator for selecting the 316 best growth model. may lead wrong conclusion. Our proposed idea apparently resolved both these issues as discussed below.

Firstly, it reduces the search space of selecting the best model by investigating only a few models. As depicted 329 in Fig. 1 and in Table 2 , 3, 4, 5 and 6, most of the models can be obtained from four models (Exponential

Model, Logistic Model, Theta-Logistic Model, Confined Exponential Model). So, an experimenter does not 331 need to compute ISRP of all the models which is itself may be a tedious computational process. Only ISRP 332 profile from these four models will give the clue for the selection of the underlying model. Thus, this work 333 further reinforces the use of ISRP and generalizes its application in model selection at a reduced effort.

The other important observation is that ISRP profile of parameter is very informative in a sense that, 335 it depicts homogeneity or heterogeneity across different growth trajectories with respect to the parameter of 336 interest, for example, whether the parameter varies or not, can be traced only in some particular time period.

If we observe the ISRP profile of r (Fig. 2b) , we notice that till time point 12, the variation are small, and the 338 same thing happens when it varies continuously (Fig. 3a, 3b) . Hence one could argue that information about 339 the parameter in the data are contained during specific time interval only. Beyond that, data points are not 340 informative as they show very high variability with respect to the parameter of interest.

In their unifying function, different choices of parameters lead to different models, thus giving a compact representation of several growth equations. It is worth mentioning that from a mathematical perspective, the 344 unifying function serves a great purpose, but from a statistical point of view it may pose difficulty in dealing 345 with real data sets as the unifying function itself is heavily parametrized by several fixed but unknown real 346 valued parameters. Thus, if we plot a network similar to Fig. 1 , the network will have only one key node (with 347 unifying function) and all other models will be leaf with no connection between them. Thus in application, 348 essentially an experimenter need to resolve a difficult estimation problem to obtain a simpler equation. On the 349 contrary, our approach in this manuscript starts with the investigation of simpler models and extrapolate to 350 complex models (if required at all). In addition the network type plot, depicted in Fig. 1 is more informative 351 than a plot with single key node, because Fig. 1 "complex to simple" strategy, then our approach is "simple to complex".

The distribution of the interval specific estimators also depends on the covariance structure of the statistical 

It is to be noted that in Fig. 3a , the values of ISRP ( r j (∆t)) with respect to time is indicative of deviation 366 from linearity at the initial phase of the growth only. This is quite natural as the impact of the growth coefficient 367 is more prominent at the early stage and as the process evolve, the information about r will be lost. Basically, 368 before the lag phase starts, logistic growth essentially behaves like exponential only. A similar idea has been 369 discussed in the context of estimating the patterns of density dependence for natural populations (Clark et al., involved in model fitting exercises. We believe that this work would be helpful for the practitioners in the field 397 of growth study. The proposed idea is verified by using simulated and real data sets from two different domains 398 (biology and marketing). We believe that this idea is unique and it contains a novel message for the scientific 399 community, in particular for applied researchers. For compact representation, we shall use matrix notations. From eqn.(4), we obtain that

In matrix notation, we have the following expression for ∇φ:

The expression of the vector of partial derivatives after evaluating at µ is given in the main text. In real variable, eqn.(6) is written as:

and η = 1 c − ζ. Taking logarithm on ζ, we obtain

Taking partial derivative with respect to x, we obtain:

So, finally we obtain that,

In real variable φ is written as:

Now taking partial derivative of φ with respect to x and y, we obtain

Using eqn. (2) , where µ t = µ 0 e rt , the distribution of r j (∆t) is given as:

The theta-logistic model is given by

and the solution X t is given by

In this model, two variables r and K and one limiting constant θ are present. Here, we only calculate the 554 variance of r and K only. The ISRP of r is given by (Pal et al., 2018) ,

In terms of real variable the function φ is given as

Now taking partial derivative of φ with respect to x, y and z, we obtain

θ , the distribution of r j (∆t) is given as:

The ISRP of K is given by (Pal et al., 2018) ,

Taking partial derivative with respect to x, we obtain:

So, finally we obtain that,

Final expressions of all the required partial derivatives are as follows:

θ , the distribution of K j (∆t) is given as:

The confined exponential model is given by

where X(t) is the population size at time t; r and K are the intrinsic growth rate and carrying capacity (asymptotic size) respectively,) and the solution X t is given by

In this model, two variables r and K are present.

In real variable φ is written as:

Now taking partial derivative of φ with respect to x, y and z, we obtain

Using eqn. (2), where µ t = K − (K − µ 0 )e −rt , the distribution of r j (∆t) is given as:

The ISRP of K is given by (Pal et al., 2018) ,

After simplification ψ is given as:

In real variable, ψ is written as:

. Taking logarithm on ζ, we obtain

Taking partial derivative with respect to x, we obtain:

So, finally we obtain that,

Using eqn. (2), where µ t = K − (K − µ 0 )e −rt , the distribution of K j (∆t) is given as:

We first discuss about the cases where after varying the parameter analytical expression of the solution of the model exists.

Appendix B.1.1. Variation in exponential growth model We start our discussion by the oldest candidate in the growth curve literature, the exponential model (Malthus, 1798) . Banks (1994) showed that after varying the parameter (r) in the exponential model (eqn. A.1), one can build connections with the normal distribution, gompertz growth, linear growth, hyperbolic growth and power law exponential model. So, we do not discuss them here. We will concentrate on the other type of variation which are not available in existing literature. 1. r = r 0 t c−1 ; c > 0 :

If we take this type of variation in the parameter r then the eqn. (A.1) turns out to Korf model (Korf, 1939) (Table 2 ; srl. 6) for which the asymptotic size tends towards ∞ and behaves like exponential model ( Fig. S1a ) and as c increases X(t) increases and goes towards asymptotic size at a much faster rate. (Table 3 ; srl.5). In this case, X(t) converges to K as t → ∞ at faster rate than in the logistic growth model (Fig. S2c) . The point of inflection remains same as logistic model.

For exponentially decaying r, eqn. (B.1) turns into a new model (Table 3 ; srl.6) in which lim t→∞ X(t) depends on c and r 0 . As c (or r 0 ) increases the asymptotic size decreases (Fig. S2d ).

5. r = r 0 e ct ; (c > 0):

For exponentially growing r, eqn. (B.1) turns into a new model (Table 3 ; srl.7) in which the asymptotic size remains K. As c increases X(t) converges to K faster than logistic model (Fig. S2e) . The point of inflection also remains the same.

6. r = r 0 1+ct ; (c > 0): For this type of variation in r, eqn. (B.1) turns into a new growth model (Table 3 ; srl.8), whose asymptotic size remains K but X(t) tends towards K very slowly (Fig. S2f ) compared to logistic growth model. For large value of c, the tendency to go towards K becomes slower. 7. r = r 0 + c sin(ωt), ( ω > 0):

In this case, eqn.(B.1) turns into a new model (Table 3 ; srl.9). For c > 0, X(t) → K in a damped oscillation manner (Fig. S2g) and also for c < 0, a similar behaviour is observed (Fig. S2h ). If we take cosine function instead of sine function in eqn.(B.1), then also we get a new model (Table 3 ; srl.10) with similar kind of behaviour (Fig. S2i, S2j) ).

2. r = r 0 θ , θ → 0 : If r takes this type of form and θ tends to zero in eqn. (A.2), then the equation turns out into gompertz model (Gompertz, 1825) . For gompertz model the carrying capacity remains K (for fixed initial size X 0 ) but point of inflection changes into K e (Fig. S3a) (Table 4 ; srl.2).

3. θ ≥ −1 :

If we consider this limitation in θ in eqn.(A.2), then it turns into Richard's growth law (Richards, 1959) . (Table 4 ; srl.5). As c increases, X(t)

goes towards K at faster rate.

6. r = r 0 (1 − ct) (c > 0):

Like the previous case, here also a new growth equation is obtained whose asymptotic size is zero (instead of K). If we increase the value of c, X(t) goes towards zero at faster rate ( Fig. S3e) (Table 4 ; srl.6).

7. r = r 0 θ t c−1 , (c > 0), θ → 0 : If we take this type of variation of r and limitation of θ in eqn. (A.2), then it turns into extended gompertz model (Bhowmick et al., 2014) for which K remains the asymptotic size and for large c, X(t) goes towards the asymptotic size at a much faster rate (Fig. S3f) (Table 4 ; srl.7). In the lower panel ( Fig. (g) , (h) and (i)), we consider r as a density dependent function to get Generalized Von-Bertalanffy Model, Crescenzo-Spina Model and Second-order Exponential Polynomial from theta-logistic Model, respectively. In all cases, r0 be the initial values of the parameter r. X0 = 10 and K = 50 are kept fixed.

For this type of density dependent variation in r and limitation in θ eqn. (A.2) transforms into Generalized Von-Bertalanffy Model (Von Bertalanffy, 1960) for which lim t→∞ X(t) = K and the rate of convergence depends on θ. For large value of θ, X(t) goes at faster rate towards K (Fig. S3g) (Table 4 ; srl.10). If we take c = 1, then generalized gompertz model turns into gompertz model (Gompertz, 1825) . If we take c = 0.5, then this model turns out into second-order exponential polynomial (Chakraborty et al., 2017) for which the asymptotic size is zero (Fig. S3h) (Table 4 ; srl.11). If we take c − 1 > 0, then generalized gompertz model turns out into Cresenzo-Spina model (Crescenzo and Spina, 2016) for which K is the asymptotic size (Fig. S3i) (Table 4; srl.12). Basically for 0 < c < 1, lim t→∞ X(t) = 0 always. Thus in generalized gompertz model, for 0 < c < 1, asymptotic size is zero and for c ≥ 1, asymptotic size is K. Fig. (a) , we consider linearly increasing function of K in Logistic model and in Fig. (b) , we consider r as a polynomial function of size in theta-logistic model. In lower panel, in Fig. (c) , we consider r as a polynomial function of X K and in Fig. (d) , we consider r as exponentially increasing function of time in theta-logistic model. For Fig. (a) , we consider r = 1 and for Fig. (b) to (d), we consider K = 100. For all cases, X0 = 10 is kept fixed.

The ISRP of r is given by

A.1) turns out to a new model (Table 2; srl. 7) whose asymptotic size is also ∞ like exponential model but X(t) tends towards it at a much faster rate than the exponential model

r = r 0 + c sin(ωt)

Table 2; srl. 8) whose asymptotic size remains ∞ but X(t) goes towards it periodically. If we keep increasing the value of r we can see much bigger period in population size (X) and also X(t) goes towards the asymptotic size at a much faster rate (Fig. S1c, S1d)

9) having similar behaviour (Fig. S1e)

This model is very sensitive with respect to the parameter choice of c. The size profile first increases and then decreases (Fig. S4b)

If we take linearly increasing time dependent function of r in eqn. (A.3), then the resulting model has asymptotic size equal to K as of the original model (Fig. S4c). However, the convergence to the asymptotic size of X(t) is much faster for large values of c

r = r 0 e −ct

then a new growth equation is obtained (Table 5; srl.6) whose asymptotic size is less than K. But lim t→∞ X(t) depends on the value of c (Fig. S4d). (c > 0) : If we take r as a inverse function of time in eqn

For c > 0, K is the asymptotic size and for large c, convergence of X(t) is slow (Fig. S4e)

For this periodic variation in r, eqn. (A.3) turns into a new model in which the asymptotic size goes towards K periodically with reduced amplitude (Fig. S4f, S4g) (Table 5; srl.8). If we take cosine function instead of sine function, then a new model

If we take linearly increasing time dependent function of K in model (A.3), then it turns out into a new model (Table 5; srl.10), for which asymptotic size is ∞ (Fig. S4j) and it goes towards it at a much faster rate for bigger

For this type of variation in K (Table 5; srl.11), it is obvious that X(t) diverges to ∞ as t → ∞ (Fig. S4k) and at faster rate than the linearly varying K

If we take exponentially decreasing time dependent function of K in model (A.3), then it turns out into a new model (Table 5; srl.12), for which asymptotic size changes into zero (Fig. S4l) and it goes towards X(t) at faster rate for large

Appendix A. Expression of the required partial derivatives for computation of variance of inter-Appendix A. 1 

Final expressions of all the required partial derivatives are as follows:In matrix notation, we have the following expression for ∇ψ:The exponential model (Malthus, 1798) is given byand the solution X t is given byIn this model, only one variable r is present and the ISRP of r is given by (Pal et al., 2018 )and η = 1 c − ζ. Taking logarithm on ζ, we 

The logistic model is given by (Verhulst, 1838)where X(t) be the population size at time t, r and K be the intrinsic growth rate and carrying capacity (asymptotic size). Banks (1994) discussed the dynamics of the logistic equation by varying K as function of time. Here, we discuss the growth behaviour of logistic equation by varying the parameter r. Apart from the connection to different existing growth equations some new models are also obtained. In the following, we categorically discuss different cases.The resulting equation (Table 3 ; srl.3) has the same asymptotic size K like logistic model (eqn. B.1). An increase in the value of r 0 (or c), X(t) tends to K at a faster rate, however, the point of inflection remains same as logistic equation (Fig. S2a ).Under this transformation, a new growth equation (Table 3 ; srl.4) is obtained. In this case growth is not monotonic. For small c, X(t) first increases and then decreases to zero. As r 0 (or c) increases X(t) → 0 at a faster rate (Fig. S2b) .

The theta-logistic model (eqn. A.2) (also referred as generalized logistic) is one of the most widely used growth equations in ecological literature. The model was first proposed by Gilpin and Ayala (1973) and other references there in). We consider both time and density dependent variation in r and K is kept as constant since almost for all the cases, the resulting growth equations do not have analytical solution. They must be solved numerically. We investigate the equations for different range of values of θ. In the following we categorically discuss each cases.

This is essentially the logistic model whose dynamics are well studied (Kot, 2001) (Table 4 ; srl.1)

In the previous section, we have discussed the growth models in which a parameter varies continuously with time following some specific functional form. It is to be noted that for each of the case, the final differential equation (after replacing r by r(t)) can be solved analytically or solutions are available using some special functions. In general this may not be the case and the differential equations must be solved numerically to obtain the size profile. In this section, we deal with few cases in which the analytical expression for the size variable X(t) is not available.If the carrying capacity increases linearly with time in eqn. (B.1), then the analytical solution for X(t)does not exists (Table 6; (Table 6 ; srl.2) with asymptotic size K. For large values of γ, X(t) converges to K at faster rate (Fig.(S5b) ) .3. r = r 0 X K c , θ > 0 :If we take r as a polynomial function of X K in eqn. (A.2), then the equation converts into Marusic-Bajzer model (Marusic and Bajzer, 1993) (Table 6; srl.3). The asymptotic size in the revised model remains K but for bigger values of c, X(t) goes towards K at slower rate ( Fig.(S5c) ) . 4. r = r 0 e ct , c > 0 :If we take r as a exponential growing function of time in eqn. (A.2), then this model turns into a new model (Table 6 ; srl.4) whose asymptotic size is K and for bigger values of c X(t) goes towards K at faster rate ( Fig.(S5d) ).