Area-concentrated search (ACS) is a simple movement rule implying that an animal searches for resources using a 'state-dependent correlated random walk'. Accordingly, a forager increases its searching intensity by reducing the directionality of movement ('intensive search mode' or ISM) when it detects a resource item, but if it searches unsuccessfully for a while, it returns to a more straight-line movement to search for new resource locations elsewhere ('extensive search mode' or ESM). We propose a modified ACS, called delayed-response ACS (dACS), which would be more efficient in resource collection than standard ACS. Instead of immediately switching from ESM to ISM when encountering a resource, as is done in standard ACS, an individual foraging in the dACS mode delays this switch by 'x' steps so it continues moving in a straight line for a while before switching to ISM. Our results show that an individual with a suitable delay parameter 'x' for the dACS achieves substantially higher foraging success than an individual with standard ACS (x=0). Optimal foraging success occurs when 'x' is approximately similar to the patch radius 'r'. This is because, with dACS, an individual can penetrate deeper into a cluster and stay longer within it, ultimately increasing the number of resources collected. Modifying the half-saturation constant 'h' also affects the success of foraging, but the effects depend on resource density and cluster size. Generally, 'h' modulates the optimal 'x' value only slightly. dACS can be interpreted as a survey movement within a resource cluster before switching from ESM to ISM. The dACS rule does not rely on complex spatial memory but only on memorizing whether resources were found or not. It may thus occur in a wide range of taxa, from organisms without a central nervous system to animals with complex brain systems. 

  For the simulations we created infinite landscapes with a single resource cluster. The cluster size and resource density varied among different scenarios. We utilized the Mat\'{e}rn Cluster Point Process using R version 3.5.3 and the ‘spatstat’ library version 1.58-2 to create resource clusters as a continuous spatial point pattern (Baddeley and Turner 2005). To understand the effects of the two landscape parameters, patch radius $r$ and resource density $u$ on foraging success (see below), we created landscapes with either the cluster radius fixed at $r=40$ and resource density $u$ set to 0.04, 0.16, or 0.64 resource items per unit area respectively, or with resource density fixed at $u=0.16$ and cluster radius varied from 20 to 40 and 80 (measured in step length). The expected number of active resource items per cluster ($\bar R$) is consequently calculated as  $\bar R=g_i^2 \pi \times u$. 

   As movement rule we implemented an area-concentrated search where an individual switches between two movement modes: intensive search mode (ISM) characterized by low directionality of movement, and extensive search mode (ESM) characterized by high directional persistence. This is referred as the standard ACS. At each time step, an individual moved one step with a constant step length ($p = 1$ spatial unit). After moving, an individual immediately harvested all resource items within its perception radius ($c=1$ spatial unit). The removed resource items were not replaced after harvesting, so that resource density in the cluster may degrade over time. We assume that an individual moved straighter, the longer the searching time ($\Delta_{S}$), i.e., the time past since the last encounter with food item (Bartoń and Hovestadt 2013; Benhamou 1992). If an individual found at least one resource item, the searching time was set to 0 ($\Delta_{S}$=0) and the ISM was initiated (see eq. \ref{eq.directionality}). Whenever an individual did not find a resource item, the value of $\Delta_{S}$ for the individual was increased to $\Delta_{S}+1$. The turning angle for the next movement step was then determined by sampling a random value from a wrapped circular normal distribution (Jammalamadaka and SenGupta 2001) with mean of zero and a standard deviation $d_{t} (\Delta_{S})$ creating from the ‘rwrappednormal’ function in R-package ‘circular’ (Version 0.5-0)(Agostinelli and Lund 2023) calculated as:

   \begin{equation}\label{eq.directionality}
   d_{t} (\Delta_{S})=d_{min}+(d_{max}-d_{min})(1-\frac{\Delta_{S}^\alpha}{(\Delta_{S}^\alpha+h^\alpha)})    
   \end{equation}   

   The minimum value of $d_{t}$ is equal to $d_{min}=0.01$ (nearly straight-line movement – ESM) when $\Delta_{S} >> h$ (the half-saturation constant), and its maximum value is equal to $d_{max}=1$ when $\Delta_{S}=0$, i.e. in case the individual just found a food item. In the latter case, the movement became highly uncorrelated (ISM) and the individual performed an area-concentrated search (see Chaianunporn and Hovestadt 2022 for more details). The shape parameter was fixed at $\alpha=3$ throughout all simulations. However, between scenarios, we varied the half-saturation constant ($h$) from 10 to 80 (see below). The half-saturation constant regulates the shift from ISM to ESM. A higher half-saturation constant means that a higher $\Delta_{S}$ value is required to switch back to the extrinsic movement mode.    

   With the standard rule for the ACS just described, a moving animal switches immediately to ISM the moment it encounters a resource item as $\Delta_S$ is set to zero at that moment. To allow for a more delayed transition from extensive to intensive searching we implement a modified version of this rule by introducing a prolonged (or cumulative) 'memory vector' of searching times over the previous $x$ time steps that affects the calculation of $\Delta_S$. More specifically, we replace $\Delta_S$ in eq. \ref{eq.directionality} by the quantity

   \begin{equation}
       \Delta_{S(x)}=\frac{\sum_{\delta=0}^{\delta=x} \Delta_\delta}{x+1}
   \end{equation}

   with $x+1$ as the total length of the memory vector (including the very last value memorized just now) and $\Delta_\delta$ as the search time memorized $\delta$ time (viz. movement) steps before the last. $\Delta_{S(x)}$ is thus simply the mean search time memorized in the last $x+1$ movement steps. The standard ACS is consequently recovered by setting $x=0$. For practical reasons, we only use even values for $x$ in our simulations, that is, we increment $x$ in steps of 2. Calculating search time as average over several past movement steps leads to a delayed switching to ISM after finding resources, the more so, the larger the value of $x$ (See Figure 1 for example). In the following, we will name this modified version of the ACS 'delayed-response area-concentrated search', abbreviated to 'dACS'. 

   For each combination of movement parameters ($x, h$; see below) we simulated movement in landscape with the following combinations of the two landscape parameters: (i) resource density ($u \in \{0.04, 0.16, 0.64 \}$ with cluster radius fixed at $r=40$), or (ii) cluster size ($r \in \{20, 40, 80\}$) with resource density fixed at $u=0.16$, that is, $3 + 4  = 7$ different landscape setting in total. To create summary statistics we replicated simulations of 40 independently individuals for each of these parameter combinations. 

   In each simulation run, a single individual was released at the southern edge of the resource cluster. The movement of animals was characterized by any combination of the two movement parameters $x \in \{0, 2, 4, ... 100 \}$ (except in $r=80$ where the maximum value of $x$ is 200) and $h \in \{10, 20, 40, 80\}$. The results presented later are consequently based on a total of 65,120 individual simulation runs ($N=40 \times 7 \times 51 \times 4 = 57120)$ plus 8,000 ($N=40 \times 50 \times 4$) extra simulations for the scenario $r=80$). At the beginning of each run, the memory vector $\Delta_{\delta=0} ... \Delta_{\delta=x}$ was initialized with the vector $500+x/2, ... 500, ... 500-x/2$ so that $\Delta_{S(x)}=500$, ensuring that individuals exhibited nearly straight-line movement (thus assuming that the individual had traveled for considerable time before reaching the resource cluster). The initial direction of the individual was randomly selected from a uniform distribution between $1/4\pi$ and $3/4\pi$. At each time step, individuals moved and searched for resources as previously described. For each individual, we performed a simulation of 1,000 movement steps. Individuals that never entered the resource cluster where removed from the data set and the simulation run was repeated.