This dissertation looked at both design-based and model-based estimation for rare and clustered populations using the idea of the ACS design. The ACS design (Thompson, 2012, p. 319) starts with an initial sample that is selected by a probability sampling method. If any of the selected units meets a pre-specified condition, its neighboring units are added to the sample and observed. If any of the added units meets the pre-specified condition, its neighboring units are further added to the sample and observed. The procedure continues until there are no more units that meet the pre-specified condition. In this dissertation, the pre-specified condition is the detection of at least one animal in a selected unit. In the design-based estimation, three estimators were proposed under three specific design setting. The first design was stratified strip ACS design that is suitable for aerial or ship surveys. This was a case study in estimating population totals of African elephants. In this case, units/quadrant were observed only once during an aerial survey. The Des Raj estimator (Raj, 1956) was modified to obtain an unbiased estimate of the population total. The design was evaluated using simulated data with different levels of rarity and clusteredness. The design was also evaluated on real data of African elephants that was obtained from an aerial census conducted in parts of Kenya and Tanzania in October (dry season) 2013. In this study, the order in which the samples were observed was maintained. Re-ordering the samples by making use of the Murthy's estimator (Murthy, 1957) can produce more efficient estimates. Hence a possible extension of this study. The computation cost resulting from the n! permutations in the Murthy's estimator however, needs to be put into consideration. The second setting was when there exists an auxiliary variable that is negatively correlated with the study variable. The Murthy's estimator (Murthy, 1964) was modified. Situations when the modified estimator is preferable was given both in theory and simulations using simulated and two real data sets. The study variable for the real data sets was the distribution and counts of oryx and wildbeest. This was obtained from an aerial census that was conducted in parts of Kenya and Tanzania in October (dry season) 2013. Temperature was the auxiliary variable for two study variables. Temperature data was obtained from R package raster. The modified estimator provided more efficient estimates with lower bias compared to the original Murthy's estimator (Murthy, 1964). The modified estimator was also more efficient compared to the modified HH and the modified HT estimators of (Thompson, 2012, p. 319). In this study, one auxiliary variable is considered. A fruitful area for future research would be to incorporate multi-auxiliary information at the estimation phase of an ACS design. This could, in principle, be done by using for instance a multivariate extension of the product estimator (Singh, 1967) or by using the generalized regression estimator (Särndal et al., 1992). The third case under design-based estimation, studied the conjoint use of the stopping rule (Gattone and Di Battista, 2011) and the use of the without replacement of clusters (Dryver and Thompson, 2007). Each of these two methods was proposed to reduce the sampling cost though the use of the stopping rule results in biased estimates. Despite this bias, the new estimator resulted in higher efficiency gain in comparison to the without replacement of cluster design. It was also more efficient compared to the stratified design which is known to reduce final sample size when networks are truncated at stratum boundaries. The above evaluation was based on simulated and real data. The real data was the distribution and counts of hartebeest, elephants and oryx obtained in the same census as above. The bias attributed by the stopping rule has not been evaluated analytically. This may not be direct since the truncated network formed depends on the initial unit sampled (Gattone et al., 2016a). This and the order of the bias however, deserves further investigation as it may help in understanding the effect of the increase in the initial sample size together with the population characteristics on the efficiency of the proposed estimator. Chapter four modeled data that was obtained using the stratified strip ACS (as described in sub-section (3.1)). This was an extension of the model of Rapley and Welsh (2008) by modeling data that was obtained from a different design, the introduction of an auxiliary variable and the use of the without replacement of clusters mechanism. Ideally, model-based estimation does not depend on the design or rather how the sample was obtained. This is however, not the case if the design is informative; such as the ACS design. In this case, the procedure that was used to obtain the sample was incorporated in the model. Both model-based and design-based simulations were conducted using artificial and real data. The study and the auxiliary variable for the real data was the distribution and counts of elephants collected during an aerial census in parts of Kenya and Tanzania in October (dry season) and April (wet season) 2013 respectively. Areas of possible future research include predicting the population total of African elephants in all parks in Kenya. This can be achieved in an economical and reliable way by using the theory of SAE. Chapter five compared the different proposed strategies using the elephant data. Again the study variable was the elephant data from October (dry season) 2013 and the auxiliary variable was the elephant data from April (wet season) 2013. The results show that the choice of particular strategy to use depends on the characteristic of the population under study and the level and the direction of the correlation between the study and the auxiliary variable (if present). One general area of the ACS design that is still behind, is the implementation of the design in the field especially on animal populations. This is partly attributed by the challenges associated with the field implementation, some of which were discussed in section 2.3. Green et al. (2010) however, provides new insights in undertaking the ACS design during an aerial survey such as how the aircraft should turn while surveying neighboring units. A key point throughout the dissertation is the reduction of cost during a survey which can be seen by the reduction in the number of units in the final sample (through the use of stopping rule, use of stratification and truncating networks at stratum boundaries) and ensuring that units are observed only once (by using the without replacement of cluster sampling technique). The cost of surveying an edge unit(s) is assumed to be low in which case the efficiency of the ACS design relative to the non-adaptive design is achieved (Thompson and Collins, 2002). This is however not the case in aerial surveys as the aircraft flies at constant speed and height (Norton-Griffiths, 1978). Hence the cost of surveying an edge unit is the same as the cost of surveying a unit that meets the condition of interest. The without replacement of cluster technique plays a greater role of reducing the cost of sampling in such surveys. Other key points that motivated the sections in the dissertation include gains in efficiency (in all sections) and practicability of the designs in the specific setting. Even though the dissertation focused on animal populations, the methods can as well be implemented in any population that is rare and clustered such as in the study of forestry, plants, pollution, minerals and so on.